The functions f(x) = (x + 1)^2 − 2 and g(x) = −(x − 2)^2 + 1 have been rewritten using the completing-the-square method. Apply your knowledge of functions in vertex form to determine if the vertex for each function is a minimum or a maximum and explain your reasoning
Answers
Answer:
f(x) — minimumg(x) — maximumStep-by-step explanation:
You want to know if the vertex is a minimum or maximum for functions ...
f(x) = (x +1)² -2g(x) = -(x -2)² +1ExtremesThe sign of the leading coefficient is the sign of the function values when x has a large magnitude.
If the leading coefficient is positive, the parabola opens upward. Hence the vertex is a minimum. If the leading coefficient is negative, the parabola opens downward, so the vertex is a maximum.
f(x)The coefficient of the (x+1)² term is +1, so the vertex is a minimum.
g(x)The coefficient of the (x-2)² term is -1, so the vertex is a maximum.
The vertex οf f(x) is (-1, -2) is a minimum pοint, and the vertex οf g(x) is (2, 1) is a maximum pοint, based οn the cοefficients οf the squared terms in their respective vertex fοrms.
What is vertex?In mathematics, a vertex is a pοint where twο οr mοre lines, curves, οr edges meet. It can alsο refer tο the highest οr lοwest pοint οn a curve οr surface.
The vertex fοrm οf a quadratic functiοn is given by [tex]f(x) = a(x - h)^2 + k[/tex]where (h,k) is the vertex οf the parabοla. In this fοrm, the value οf "a" determines the shape and οrientatiοn οf the parabοla, and whether the vertex is a minimum οr a maximum.
Let's first rewrite the given functiοns in vertex fοrm:
[tex]f(x) = (x + 1)^2 - 2= 1(x - (-1))^2 - 2= 1(x - (-1))^2 + (-2)[/tex]
Therefore, the vertex of f(x) is (-1, -2), and since the coefficient of the squared term is positive, the parabola opens upward. This means that the vertex (-1, -2) is a minimum point.
[tex]g(x) = -(x-2)^2+1[/tex]
[tex]= -1(x - 2)^2 + 1[/tex]
Therefore, the vertex of g(x) is (2, 1), and since the coefficient of the squared term is negative, the parabola opens downward. This means that the vertex (2, 1) is a maximum point.
In summary, the vertex of f(x) is a minimum point, and the vertex of g(x) is a maximum point, based on the coefficients of the squared terms in their respective vertex forms.
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Related Questions
Need help, will give points.
Answers
The value of (f o g)(3) using composite function is 35
What is composite function?Composite function refers to the combining of functions in a manner where the output from one function becomes the input for the next function.
To calculate the composite function (f o g)(3), we use the techniques below.
From the question,
Given:
f(x) = 3x-1g(x) = x²+3Note that,
(f o g)(3) = f(g)(3)Step 1: Find g(3)
To find g(3) we suebtitute the value of x in g(x) by 3Therefore,
g(3) = 3²+3g(3) = 12Step 2: To find f(g)(3), we subetitute the value of x in f(x) by 12
f(g)(3) = 3(12)-1 f(g)(3) = 36-1f(g)(3) = 35Hence, (f o g)(3) is 35
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in an examination held it was discovered that 960 candidate took mathematics 780 took physics and 572 took chemistry .of these numbers 300 took both Mathematics and physics 340 took both Mathematics and chemistry 260 took both physics and chemistry while 130 candidates took all the three papers .how many candidates were there in that centre ?how many took Mathematics and physics? Only translate in a venn diagram
Answers
Answer 1452
Step-by-step explanation:
n(m n p n c) =n(m) + n(p) + n(c) - n(m n p) - n(m n c) - n(p n c) + n(m n p n c) = 960+ 780 + 572 - 300 - 340 - 260 + 130 = 1452
A rectangular room is 2 times as long as it is wide, and its perimeter is 48 meters. Find the dimensions of the room.
Answers
Answer: Let's assume that the width of the rectangular room is "x" meters. Then, according to the problem, the length of the room is 2 times the width, which means the length is "2x" meters.
The perimeter of the room is the sum of all four sides, so we can write:
Perimeter = 2 × (length + width)
Substituting the values, we get:
48 = 2 × (2x + x)
Simplifying the expression, we get:
48 = 2 × 3x
Dividing both sides by 2, we get:
24 = 3x
Solving for "x", we get:
x = 8
Therefore, the width of the room is 8 meters and the length is 2 times the width, which is 2 × 8 = 16 meters.
So the dimensions of the room are 8 meters by 16 meters.
Step-by-step explanation:
1. Explain the difference between an in-text citation and the reference list.
Answers
Answer:
In-text citations are the short citations you include in the written text that help a reader to understand which sources you are quoting or referring to in your writing example(Field, 2022). Reference is the full details of the source you have cited in your writing.
A surveyor is measuring the width of a pond. She chooses a landmark on the opposite side of the pond, and measures the angle to this landmark from a point 50 feet away from the original point. How wide is the pond to the nearest ft? A B 50 ft C
Answers
Answer: This issue can be resolved by employing trigonometry. Let's say that the surveyor is standing at point A, and the landmark on the other side of the pond is at point B. We can also make the assumption that the distance between points A and B is the width of the pond we need to find. Now, let's draw a line from point A to point B to represent the pond's width. Let's call this distance x. The surveyor's distance from the pond can also be represented by drawing a line 50 feet away from point A to point C.
The tangent of an angle is the opposite side (in this case, x) divided by the adjacent side (in this case, 50 feet), so we can now use the tangent function to determine x. We now have:
Rearranging this equation allows us to solve for x: tan(theta) = x/50, where theta is the angle between the lines from point A to point B and point A to point C.
We do not know the value of theta, but we are aware that the surveyor measured the angle to the landmark. x = 50*tan(theta) We should expect that the assessor estimated the point in degrees. Then, we can multiply the angle's tangent by 50 to get x using a calculator. For instance, if the surveyor measured an angle of 30 degrees, we would have:
The pond's width is therefore approximately 28.85 feet, rounded to the nearest foot, as x = 50tan(30) = 500.577. The response isn't given in the decisions, yet it is nearest to 29 feet
Step-by-step explanation:
Riley was a spectator at his town's air guitar competition. Contestants were allowed to play either the acoustic or electric air guitar, but not both. Riley recorded which type of guitar each contestant played. He also counted the number of contestants wearing different kinds of pants, as there were some interesting stylistic choices.
What is the conditional probability that a randomly selected contestant played an acoustic guitar, given they wore leather pants?
show all steps
Answers
The conditional probability that a randomly selected contestant played an acoustic guitar, given they wore leather pants, is 0.6 or 60%.
What is Probability ?
Probability is the measure of the likelihood or chance of an event occurring. It is a mathematical concept that is used to quantify uncertainty and provide a basis for making informed decisions in various fields such as science, engineering, economics, and finance.
To find the conditional probability that a randomly selected contestant played an acoustic guitar, given they wore leather pants, we need to use the formula for conditional probability:
P(A | B) = P(A and B) / P(B)
where A is the event that the contestant played an acoustic guitar, and B is the event that the contestant wore leather pants.
We are not given the total number of contestants in the competition, but we can use the information provided in the table to calculate the probabilities of A and B, as well as the joint probability of A and B.
Let's first calculate the probability of B:
P(B) = number of contestants wearing leather pants / total number of contestants
P(B) = 15 / 45
P(B) = 1/3
Now, let's calculate the joint probability of A and B, i.e., the probability that a randomly selected contestant played an acoustic guitar and wore leather pants:
P(A and B) = number of contestants who played acoustic guitar and wore leather pants / total number of contestants
P(A and B) = 6 / 45
Finally, we can use the formula for conditional probability to find the probability that a randomly selected contestant played an acoustic guitar, given they wore leather pants:
P(A | B) = P(A and B) / P(B)
P(A | B) = (6 / 45) / (1/3)
P(A | B) = 0.6
Therefore, the conditional probability that a randomly selected contestant played an acoustic guitar, given they wore leather pants, is 0.6 or 60%.
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A box contain 12 items of which 3 are defective.a sample of 3 items is selected at random from this box if x represents the number of defective items in the 3 selected items describe the random variable x
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The random variable x can take on the values of 0, 1, 2, or 3, representing the number of defective items in the sample of 3 items.
Describing the random variable xThe random variable x represents the number of defective items in a sample of 3 items selected at random from the box.
Since there are 3 defective items out of 12 total items, the probability of selecting a defective item on the first draw is 3/12 or 1/4.
After the first draw, there are only 11 items remaining in the box, of which 2 are defective. Therefore, the probability of selecting a defective item on the second draw, given that the first draw was also defective, is 2/11.
Similarly, the probability of selecting a defective item on the third draw, given that the first two draws were also defective, is 1/10.
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help timed asap hurry
Answers
The value of y for the given quadrilateral is 5.
What is a quadrilateral?A closed quadrilateral has four sides, four vertices, and four angles. It is a form of polygon. In order to create it, four non-collinear points are joined. Quadrilaterals always have a total internal angle of 360 degrees.
The Latin words quadra, which means four, and latus, which means sides, are combined to form the English term quadrilateral. A quadrilateral does not always have to have equal lengths on each of its four sides. As a result, depending on the sides and angles, we might have several sorts of quadrilaterals.
We know that the opposite sides of the quadrilateral have a sum of 180 degrees.
Thus,
23y + 13y = 180
36y = 180
y = 5
Hence, the value of y for the given quadrilateral is 5.
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Part 1 of 3
O Points: 0 of 1
A manufacturing company makes two types of water skis, a trick ski and a slalom ski. The relevant manufacturing
data are given in the table.
Labor-Hours per Ski
Trick Ski
9
1
Department
Fabricating
Finishing
Answer parts (A), (B), and (C) below.
Slalom Ski
6
1
Save
Maximum Labor-Hours
Available per Day
432
63
(A) If the profit on a trick ski is $60 and the profit on a slalom ski is $50, how many of each type of ski should be
manufactured each day to realize a maximum profit? What is the maximum profit?
The maximum profit is $. The maximum occurs when
trick skis and
slalom skis are produced.
Answers
The maximum profit of manufacturing company is = $3150
What do you mean by the term Linear Equation in one variable ?
If an equation is in ax + b = c and there is only one variable in the equation then it is called linear equation in one variable. a , b , c are real number and a is not equal to 0.
To maximize profit, we must determine the number of trick and slalom skis to produce each day, subject to the constraints of available labor hours.
Let x be the number of skis and y be the number of slalom skis produced per day.
The objective function to be maximized is:
Profit = 60x50 a
With the following limitations:
Production hours: 9x 6 a ≤ 432
Final hours: x y ≤ 63
Non-negativity constraints: x ≥ 0, y ≥ 0
We can solve this linear programming problem using graphical or algebraic methods. Here we use algebraic methods:
From the second limit, we get y ≤ 63 - x.
If this inequality is replaced by the first limit, we get:
9 x 6 (63 - x) ≤ 432
3x ≤ 54
x ≤ 18
This means that the maximum production rate is 18 pieces per day. Replacing x = 18 by the inequality y ≤ 63 - x, we get:
y ≤ 63 – 18 = 45
So the maximum production is 45 slalom skis per day.
To verify that this is the correct solution, we need to ensure that it satisfies the non-negativity constraints:
x ≥ 0 and y ≥ 0 are already given in the assignment statement.
Therefore, the maximum profit is:
Profit = 60(18) 50(45) = $3150
The biggest profit is when 18 trick skis and 45 slalom skis are made per day.
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Find the value of J.
Answers
The value of j is equal to 9√3 meters.
What is a 30-60-90 triangle?In Mathematics and Geometry, a 30-60-90 triangle is also referred to as a special right-angled triangle and it can be defined as a type of right-angled triangle whose angles are in the ratio 1:2:3 and the sides are in the ratio 1:√3:2.
This ultimately implies that, the length of the hypotenuse of a 30-60-90 triangle is double (twice) the length of the shorter leg (adjacent side), and the length of the longer leg (opposite side) of a 30-60-90 triangle is √3 times the length of the shorter leg (adjacent side).
Note: j is the side opposite to the 60° angle.
The side opposite to the 60° angle, j = 9 × √3
The side opposite to the 60° angle, c = 9√3 meters.
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What is the answer to this problem?
Answers
Thus, the event "Paulina gets injured playing soccer" is more likely to occur than the event "Paulina gets injured playing volleyball."
Answer:
C) neither. Both events are equally likely.
Step-by-step explanation:
Please help thank you very much!
Answers
Hello!
Lets consider what the question asks for:
--> graphs of either line intersecting/parallel/identical
--> # of solutions the system has
The information given:
6x + y = 25
x + 3y = 3
Let's solve the FIRST problem
--> graphs of either line intersecting/parallel/identical
--> let's put both equations into slope-intercept form, where we
isolate the y-variable all by itself to one side
[tex]6x+3y=25\\3y=-6x+25\\\\y=-2x+\dfrac{25}{3}[/tex]
[tex]x+3y=3\\3y=-x+3\\y=-\dfrac{1}{3}x+1[/tex]
--> both equations AREN'T parallel as the slope (coefficient of x-
variable) aren't equal
--> both equations AREN'T identical as they don't look the same
--> both equations ARE INTERSECTING as they have different
slopes and aren't identical
Let's solve our second question:
--> how many solutions does the system have:
--> both lines intersect
-->both lines are linear as the highest power that the x-variable
has are 1
--> thus there is only ONE SOLUTION
Answer:
Intersecting
Has ONE SOLUTION
convert to slope-intercept form (y=mx+b)
Answers
-3y = -5x + 15
Now divide completely isolate y by dividing -3 from both sides:
y = (-5/-3)x + (15/-3)
Simplify:
y = 5/3 x + (-5)
This becomes
y = 5/3 x - 5
or
y = 5x/3 - 5
IM BEGGING U SOMEONE
Answers
The dilation of the black figure to blue figure is a reduction with a scale factor of 0.5
How to find the scale of dilation?A dilation is a transformation that produces an image that is the same shape as the original, but is a different size.
It can either be an enlargement or a reduction.
An enlargement means a scale factor greater than 1 while a reduction is a scale factor less than 1.
Thus, this dilation is a reduction.
Coordinates of initial figure are:
(-8, 10), (10, 10), (8, -8), (2, -8)
Coordinates of dilated figure are:
(1, -4), (4, -4), (-4, 5), (5, 5)
Thus, it is clear that the scale factor is 0.5
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As a Mathematics teacher, list and explain the intervention strategies you would
apply in your teaching in assisting a child who is like Peter and why do you think your
strategies will be effective? Substantiate your response by
Answers
Answer:
1. Differentiated Instruction: This intervention involves tailoring lessons and activities to meet the individual learning needs and preferences of students. For example, a teacher can design various instructional materials for students with different skill levels to ensure they engage in the lesson and learn effectively.
2. Explicit Instruction: This intervention involves breaking down a complex process or concept into smaller, more manageable parts, and explaining each part clearly and explicitly. The teacher provides ample opportunities for students to practice and apply what they have learned.
3. Formative Assessment: This intervention involves using ongoing, informal assessments to monitor students' progress and provide feedback to both students and teachers to make necessary adjustments. Teachers can use formative assessments to identify areas of difficulty and intervene early.
4. Collaborative Learning: This intervention involves encouraging students to work together in pairs or small groups to complete tasks or solve problems. Within these groups, students share ideas, clarify misunderstandings, and learn from each other.
Differentiated Instruction, Small Group Instruction, Formative Assessment, Peer Tutoring, Real-World Connections and Positive Feedback are the strategies to assist the child.
What is Teaching?Teaching can be defined as engagement with learners to enable their understanding and application of knowledge, concepts and processes
Differentiated Instruction, Small Group Instruction, Formative Assessment, Peer Tutoring, Real-World Connections and Positive Feedback:
These strategies are effective because they are research-based and have been shown to support students' learning in mathematics.
By catering to the specific needs of students, providing individualized support, and making connections to the real world, students are more likely to engage in mathematics, understand mathematical concepts, and develop positive attitudes towards mathematics.
Hence, Differentiated Instruction, Small Group Instruction, Formative Assessment, Peer Tutoring, Real-World Connections and Positive Feedback are the strategies to assist the child.
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Translateeachexpression. Words
Expression
1. 2. 3. 4.
“five more than a number” “the quotient of a number and -2” “twice a number”
“a number decreased by 7”
“the product of a number and twelve” “three subtracted from a number” “earning $15 per hour”
“$5 less than the cost of the ticket”
“the sum of a number and 24”
“50 markers divided among several students” “two-fifths of a number”
“half a number minus nineteen”
“11 less than the product of a number and -4”
“the total of eight times a number and five”
“the difference of a number squared and 14”
“three-fourths of a number plus17
Answers
Answer:
Step-by-step explanation:
5 + x (where x represents the number)
x / (-2)
2x
x - 7
12x
x - 3
$15h (where h represents the number of hours worked)
C - $5 (where C represents the cost of the ticket)
x + 24
50 / n (where n represents the number of students)
(2/5)x
(1/2)x - 19
-4x - 11
8x + 5
x^2 - 14
(3/4)x + 17
----------------------------------
Five added to a number
The result of dividing a number by -2
Two times a number
A number decreased by 7
The result of multiplying a number by 12
A number decreased by 3
Earning $15 per hour
The cost of the ticket minus $5
The sum of a number and 24
Dividing 50 markers among several students
Two-fifths times a number
Half of a number minus 19
The result of subtracting 11 from the product of a number and -4
The sum of eight times a number and five
The difference between the square of a number and 14
Three-fourths times a number plus 17
Multiply and write the fraction or mixed number in simplest form 2/5x6/7=
Answers
Answer:
[tex]\frac{12}{35}[/tex]
Step-by-step explanation:
[tex]\frac{2}{5}[/tex] x [tex]\frac{6}{7}[/tex] = [tex]\frac{12}{35}[/tex]
[tex]\frac{12}{35}[/tex] is already in simplest form, so the answer is [tex]\frac{12}{35}[/tex]
Describe the graph of the equation x=4. Is the equation a function?
Answers
Answer:
x = 4 is a straight, vertical line at 4 on the x axis.
What is the surface area of the triangular prism shown if a = 20 units, b = 29 units, c = 22 units, w = 15 units, and h = 21 units?
Answers
The surface area of the triangular prism is 1554 square units
Step-by-step explanation:
To find the surface area of a triangular prism, we need to add up the areas of all its faces. A triangular prism has two congruent triangular bases and three rectangular faces.
First, let's calculate the area of each triangular base. We can use the formula for the area of a triangle, which is A = 1/2 * base * height.
The base of each triangle is one of the sides of the triangle, and the height is the height of the prism, which is h = 21 units. We have:
Area of one triangular base = 1/2 * b * h = 1/2 * 29 * 21 = 304.5 square units
Since there are two congruent triangular bases, the total area of the bases is:
Total area of the bases = 2 * 304.5 = 609 square units
Next, let's calculate the area of each rectangular face. We can use the formula for the area of a rectangle, which is A = length * width.
The length of each rectangular face is one of the sides of the triangle, and the width is the height of the prism, which is h = 21 units. We have:
Area of one rectangular face = w * h = 15 * 21 = 315 square units
Since there are three rectangular faces, the total area of the rectangular faces is:
Total area of the rectangular faces = 3 * 315 = 945 square units
Finally, we can add up the areas of the bases and the rectangular faces to find the total surface area of the prism:
Total surface area = Total area of the bases + Total area of the rectangular faces
= 609 + 945
= 1554 square units
Therefore, the surface area of the triangular prism is 1554 square units.
Help
Show your steps!!!!!!!! THANK YOU
Answers
The equation that represents the represents the rational function in the graph, based on the asymptotes and the intercepts is; y = ((3/4)·x - 3)/(x + 3)
What is an asymptote of a function?An asymptote is a line that approaches a curve but at any infinite distance, is still at some distance from the curve.
The asymptotes of the functions are;
Horizontal asymptote; x = -3
Vertical asymptote; y = 1
The points on the graph are;
y-intercept = (0, -1)
x-intercept = (4, 0)
The function is a rational function of the form f(x) = P(x)/Q(x)
The horizontal asymptote of x = -3 indicates that the denominator has a factor of (x + 3)
The vertical asymptote of y = 1 indicates that the degree of the numerator and the denominator are the same and the leading coefficients are the same
The function is therefore in the form; f(x) = (a·x + c)/(x + 3)
a = 1, therefore;
f(x) = (x + c)/(x + 3)
The y-intercept (0, -1) indicates that when x = 0, y = -1, therefore;
-1 = (0 + c)/(0 + 3)
-3 = c
c = -3
The function is therefore;
f(x) = y = (a·x - 3)/(x + 3)
The x-intercept (4, 0), indicates that when y = 0, x = 4, therefore;
0 = (a·4 - 3)/(4 + 3)
a·4 - 3 = 0
a = 3/4 = 0.75
The function is therefore;
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We have been given that the ages of the winners of a cycling tournament are approximately bell-shaped. The mean age is 28.5 years, with a standard deviation of 3.5 years.
Answers
The z-score for an age of 30 years is 0.43.
What is confidence interval?A confidence interval (CI) for an unknown parameter in frequentist statistics is a range of estimations. The most popular confidence level is 95%, but other levels, such 90% or 99%, are occasionally used for computing confidence intervals. The fraction of related CIs over the long run that actually contain the parameter's true value is what is meant by the confidence level.
To convert an age to a z-score, we use the formula:
z = (x - μ) / σ
where:
x = the age we want to convert to a z-score
μ = the mean age
σ = the standard deviation of the ages
For example, if we want to convert an age of 30 years to a z-score using the given information with μ = 28.5 and σ = 3.5, we have:
z = (30 - 28.5) / 3.5
z = 0.43
So the z-score for an age of 30 years is 0.43.
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Complete question:
after allowing 20% discount on the marked price of a smart boad, 15% vat was levied and sold it . If the different between the selling price with vat and selling price after discount is rs 4500, find the marked price of that smart board.
Answers
Answer:
mp = 37500
Step-by-step explanation:
let mp be x
D.A = 20×x/100
= 20x/100
=x/5
price after discount = mp - DA
= x-x/5 = 4x/5
VAT= 15%of 4x/5
= 15/100 × 4x/5
= 60x/500
sp with vat= 4x/5 +60x/500
= 460x/500
sp with vat - sp after discount = 4500
460x/500-4x/5=4500
60x= 4500×500
x= 37500
write an equation for the ellipse with vertices (-8, 5), (4, 5) and foci (-7,5),(3,5)
Answers
Answer:
Step-by-step explanation:
16
lbozo
If the median of the following given data is 19, find the value of P उमेर वर्षमा (Age in Year) 12-18 18-24 24-30 feneffen (No. of Students) 10 P 4 6-12 4 30-36 3 36-42 3
Answers
If the median of the given data is 19, the value of P is 6.
What is median?In statistics, the median is a measure of central tendency that represents the middle value of a data set when it is arranged in numerical order. It is the value that separates the upper half of the data set from the lower half. To find the median of a set of data, the data must first be arranged in order from lowest to highest (or highest to lowest). If there is an odd number of data points, the median is simply the middle value.
Here,
To find the value of P, we first need to find the median class interval. We know that the median is 19, which means that half the data is above 19 and half the data is below 19. We can calculate the cumulative frequency of the data by adding up the frequencies of each class interval.
Age f cf
12-18 10 10
18-24 P 10+P
24-30 4 14+P
30-36 6 20+P
36-42 3 23+P
The total frequency is 23 + P. The median class interval is the interval that contains the 12th value (since we have 23 + P total values). To find the median class interval, we need to find the cumulative frequency that is closest to, but not less than, 12. In this case, the cumulative frequency that is closest to 12 is 10, which corresponds to the first class interval (12-18). Therefore, the median class interval is 12-18. We can use the following formula to find the value of P:
Median = L + (n/2 - F) * w
where L is the lower limit of the median class interval, n is the total frequency, F is the cumulative frequency up to the median class interval, and w is the width of the class interval.
For the median class interval of 12-18:
L = 12
n = 23 + P
F = 10
w = 6
Substituting these values into the formula, we get:
19 = 12 + ((23+P)/2 - 10) * 6
Simplifying the equation:
7 = ((23+P)/2 - 10)
14 = 23+P - 20
P = 6
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sherita is painting the hallway of her apartment she draws a graph illustrating the proportional relationship of the minutes she works x to the passes through the point(16,18)
Part A: which ordered pair on the line would display Sherita’s unit rate of square feet painted per minute.
Part B:select all of the points that lie on the line
(2,2.5)
(3.5,3.9375)
(4,5)
(32,36)
(10,11.5)
(0,0)
Answers
The answer of the given question based on graph is Part A: the unit rate of square feet painted per minute is 1.125, so the ordered pair that displays this unit rate is (1, 1.125) and Part B: the points that lie on the line are (3.5,3.9375), (4,5), and (32,36).
What is Slope?Slope refers to the steepness of a line or a surface. More specifically, slope is defined as the ratio of the vertical change (rise) between two points to the horizontal change (run) between the same two points on a line or a surface. In other words, slope measures how much a line or a surface rises or falls as it moves horizontally.
Since the relationship between the minutes worked and the square feet painted is proportional, we can use the slope-intercept form of a linear equation to represent it:
To find the equation of the line, we can use the given point (16,18) and the fact that the relationship is proportional. If we let the unit rate be r (in square feet per minute), then we have:
18 = r(16)
r = 18/16
r = 1.125
So the equation of the line is:
y = 1.125x
Part A: The unit rate of square feet painted per minute is 1.125, so the ordered pair that displays this unit rate is (1, 1.125), since the slope represents the unit rate in a proportional relationship.
Part B: To determine which of the points lie on the line, we can plug in the x-values into the equation y = 1.125x and see if the resulting y-values match.
(y - 18) / (x - 16) = slope
Using the slope calculated above, we can simplify this equation to:
y = 1.125(x - 16) + 18
Now, we can plug in the x and y values for each point and check if they satisfy this equation:
(2,2.5): 2.5 = 1.125(2 - 16) + 18, not on the line
(3.5,3.9375): 3.9375 = 1.125(3.5 - 16) + 18, on the line
(4,5): 5 = 1.125(4 - 16) + 18, on the line
(32,36): 36 = 1.125(32 - 16) + 18, on the line
(10,11.5): 11.5 = 1.125(10 - 16) + 18, not on the line
(0,0): 0 = 1.125(0 - 16) + 18, not on the line
Therefore, the points that lie on the line are (3.5,3.9375), (4,5), and (32,36).
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Find the slope of the line shown in the graph. On a coordinate plane, a line goes through (0, negative 1) and (3, 0). a. 3 c. Negative one-third b. -1 d. One-third Please select the best answer from the choices provided
Answers
Answer:
The slope is [tex]\frac{-1}{3}[/tex]
Step-by-step explanation:
From the two points (0,-1) (3,0). The y values are 0 and -1. The x values are 3 and 0. The slope is the change in y over the change in x. You find the change by subtracting.
[tex]\frac{0-1}{3-0}[/tex] = [tex]\frac{-1}{3}[/tex]
Helping in the name of Jesus.
Section 1: Problem Statement - Finance 302
John Smith is 30 years old and graduated from CSUSM some years back, with a Business degree and an emphasis in Marketing. John is currently employed as a Marketing Manager at a well-known corporation. He has progressed well in his career, with the ultimate goal of becoming the company’s CEO. John’s current salary of $78,000 has increased at an average rate of 5% per year, with routine merit raises, and he expects it keep increasing.
John’s firm, ABC Corporation, has a defined contribution plan (401k) plan in place. Employees are allowed to contribute up to 15% of their gross annual salary pre-tax. Unfortunately, John has not yet taken his professor’s advice to “Save, Start Young, and Pay Yourself First.” Instead, John has enjoyed his post-college, nice-salary life by leasing a new car, renting an apartment and going out to Players every weekend. Now that he has wedding plans on the horizon, John has come to the realization (with help from his fiancée, Jane Doe) that it’s time to start saving while he’s still relatively young!
John expects that the lovebirds’ two largest future expenses will be the cost of a wedding (short-term), then later the down payment on a house (intermediate-term). The couple plans to spend $10,000 of their own money on the wedding in twelve months. They also hope to purchase a $400,000 house within 5 years. Jane’s parents have promised to match their 10% down payment on the house, but only if they manage to save it within 5 years. Finally, John would like to retire at age 60, since his own father died at age 59 and did not get to enjoy the fruits of his labor in retirement. Both future spouses agree that John will automate his savings by setting up monthly contributions to his wedding fund, house fund and 401k accounts.
Section 2: Questions
Your assignment should begin with a “Data” section, documenting all of the numerical assumptions provided in this assignment.
Your calculations will require the use of a financial calculator. Please provide your detailed calculations, step-by-step, including calculator functions, in order to receive maximum credit – or partial credit if the final answer is incorrect.
1) John Smith is finally ready to take advantage of his employer’s 401k plan towards his retirement goal of age 60. Assume that John contributes 10% of his current salary every year, with an 8% annual return compounded annually. How much would he have saved at age 60?
2) a) How much money would John have to save every year to achieve an age 60 balance of one million dollars?
b) What percentage of his current salary is that annual savings amount?
3) John’s fiancée, Jane Doe, is adamant about getting married in the next year. She is insisting that John makes saving towards the $10,000 needed their top priority. John recalls that his professor said, “don’t invest in long-term investments with short-term money.” Therefore, he plans to keep the wedding account in the bank and buy short-term (under 1-year maturity) CD’s. Assuming John stays continuously invested in CD’s yielding 2% annual yield for the duration of each monthly deposit from the beginning month (Month 0), how much will he have to contribute to the wedding fund every month for the next 12 months?
4) Jane does acknowledge that saving for a home down payment is not as big of a priority. But both future spouses agree that they should start putting money away towards the goal of $40,000 within five years. John recalls that an intermediate-term savings plan should not take as much risk, so he will plan to earn 4% annually with a conservative strategy for their house fund. How much money will he have to save every month for the next 60 months to accumulate $40,000?
5) Planning on an early retirement at age 60, John will start withdrawing from his 401k every month. He plans to start with $1,000,000 in his 401k, with a life expectancy of 85 years. Assuming a rate of return on his account of 6% annually, how much can he withdraw every month for his retirement expenses? (HINT: use I/Y = 6%/12 = 0.5% monthly)
Answers
Data: John's current pay is $78,000. Annual salary increase rate: 5% Maximum 401k contribution: 15% of pre-tax gross yearly salary Every year, John intends to pay 10% of his current earnings to his 401k.
Annual compounded return on 401k investments is expected to be 8%.
$10,000 wedding money target in 12 months
Short-term CD interest rate: 2%
Goal for house down payment: $40,000 in 5 years
Annual expected return on housing fund investments: 4%
At the age of 60, your 401k balance is $1,000,000.
85 years is the average life expectancy.
0.5% (6%/12) monthly rate of return on retirement investments
To figure out John's 401k balance at age 60, do the following:
Compute John's annual contribution to his 401k: $78,000 x 10% = $7,800
Employ the financial calculator's future value (FV) function:
N (number) = 30 (number of years until retirement)
I/Y = 8% (annual rate of return)
PMT = -$7,800 (negative because it’s a cash outflow)
PV = 0 (no initial balance)
FV = $?
FV = $1,057,323.95
Therefore, John would have saved approximately $1,057,323.95 at age 60.
a) To achieve an age 60 balance of one million dollars:
Use the present value (PV) function on the financial calculator:
N = 30 (number of years until retirement)
I/Y = 8% (annual rate of return)
PMT = -$? (negative because it’s a cash outflow)
PV = 0 (no initial balance)
FV = $1,000,000
PMT = -$9,308.79
Therefore, John would have to save approximately $9,308.79 every year to achieve an age 60 balance of one million dollars.
b) To find the percentage of his current salary that the annual savings amount represents: Divide the annual savings amount by John’s current salary: $9,308.79 / $78,000 = 0.1194
Multiply by 100 to get the percentage: 0.1194 x 100 = 11.94%
Therefore, the annual savings amount represents approximately 11.94% of John’s current salary.
To calculate how much John will have to contribute to the wedding fund every month for the next 12 months:
Use the present value (PV) function on the financial calculator:
N = 12 (number of monthly deposits)
I/Y = 2%/12 = 0.1667% (monthly rate of return)
PMT = -$? (negative because it’s a cash outflow)
PV = 0 (no initial balance)
FV = $10,000
PMT = -$821.47 As a result, John will have to contribute $821.47 to the wedding fund each month for the following 12 months.
To determine how much John must save each month for the next 60 months in order to acquire $40,000:
Employ the financial calculator's future value (FV) function:
N = 60 (number of months) (number of months)
I/Y = 4%/12 = 0.3333% (monthly interest rate)
PMT = -$? (negative since it represents a monetary outflow)
PV = 0 (no starting balance) (no initial balance)
FV = $40,000
PMT = -$603.94
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Find the distance between the following pairs of points.
(2,-1),(7,-1)
Answers
Answer:
5
Step-by-step explanation:
[tex]\sqrt{(7-2)^2+((-1)-(-1))^2\\[/tex]
subtract 2 from 7
[tex]\sqrt{5^2+((-1)-(-1))^2[/tex]
raise 5 to the power of 2
[tex]\sqrt{25+((-1)-(-1))^2[/tex]
Multiply -1 by -1
[tex]\sqrt{25+(-1+1)^2[/tex]
Add -1 and 1
[tex]\sqrt{25+0^2[/tex]
Add 25 and 0
[tex]\sqrt{25[/tex]
rewrite 25 as 5^2
[tex]\sqrt{5^{2}[/tex]
pull term out from under the radical assuming positive real numbers.
5
A rectangular skating rink is 30 m by 20 m. It is to be doubled in area by adding a strip at one end and a strip of equal width along one side. Find the width of the strip.
Answers
The width of the added strip is 10 meters.
What is Area?
Area is a measure of the amount of surface enclosed by a two-dimensional shape or figure. It is usually expressed in square units, such as square meters or square inches, and is calculated by multiplying the length of a side or a diameter by itself or by using an appropriate formula for the shape.
If the area of the skating rink is doubled by adding a strip of equal width along one side and another strip at one end, then the new dimensions of the skating rink would be
=> (30 + x) × (20 + x),
where x is the width of the added strip.
According to the problem, the new area of the skating rink is double the original area, so we can set up the following equation:
=> (30 + x)(20 + x) = 2(30 ×20)
Expanding the left-hand side and simplifying, we get:
=> 600 + 50x + x² = 1200
Rearranging and solving for x, we get:
=> x² + 50x - 600 = 0
Factoring the left-hand side, we get:
=> (x + 60)(x - 10) = 0
Therefore, the possible values for x are x = -60 and x = 10. Since the width of the strip cannot be negative, we reject x = -60 and conclude that the width of the added strip is 10 meters.
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a rectangular park has a length of (x + 5) and a width of (x - 2). What is the area of the park?
Answers
The area of a rectangle is given by the formula:
Area = length x width
In this case, the length of the park is x + 5 and the width is x - 2. So, we can substitute these expressions into the formula to get:
Area = (x + 5) x (x - 2)
Expanding the product, we get:
Area = x^2 + 5x - 2x - 10
Simplifying, we have:
Area = x^2 + 3x - 10
Therefore, the area of the rectangular park is given by the expression x^2 + 3x - 10.