In 1981 number of ways a committee of five members of the department if at least one woman must be on the committee.
By choosing some items from a set and creating subsets, permutation and combination are two approaches to express a collection of things. It outlines the numerous configurations for a certain set of data. Permutations are the selection of data or objects from a set, whereas combinations are the order in which they are represented. Both ideas are critical to mathematics.
The number of ways of picking 5 from 14 is what the question is actualy asking minus combinations of 5 from 7 , because there must be 1 woman
so 5 from 14 is given by :
= [tex]\frac{14*13*12*11*10}{5*4*3*2*1}[/tex]
which is :
14 x 13 x 11 = 2002
combinations of 5 from 7 is :
(9x8x7x6x5) / (5x4x3x2x1)
[tex]\frac{7*6*5*4*3}{5*4*3*2*1}[/tex]
which is :
7 x 3= 21
so the final answer is 2002 - 21 = 1981.
Therefore, in 1981 ways a committee of five members of the department if at least one woman must be on the committee.
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What would cause a discontinuity on a rational function (a polynomial divided by another polynomial)?
The function has a horizontal asymptote at y = 3. Other types of discontinuities can also occur in rational functions
What are polynomials ?A polynomial is a mathematical expression that consists of variables and coefficients, which are combined using arithmetic operations such as addition, subtraction, multiplication, and non-negative integer exponents.
A rational function can have a discontinuity at any point where the denominator of the function becomes zero since division by zero is undefined. These points are called "vertical asymptotes."
For example, consider the rational function f(x) = (x² - 1) / (x - 1). The denominator becomes zero when x = 1, which causes a vertical asymptote at x = 1. At x = 1, the function approaches positive infinity from the left-hand side and negative infinity from the right-hand side. This creates a "hole" or a "removable discontinuity" in the graph of the function.
Another type of discontinuity that can occur in a rational function is a "horizontal asymptote." This occurs when the degree of the numerator is less than the degree of the denominator. In this case, the function approaches a horizontal line (the horizontal asymptote) as x approaches infinity or negative infinity.
For example, consider the rational function f(x) = (3x² - 2x + 1) / (x² + 1). As x approaches infinity or negative infinity, the function approaches the horizontal line y = 3.
Therefore, the function has a horizontal asymptote at y = 3.
Other types of discontinuities can also occur in rational functions, such as "slant asymptotes" or "oscillating behavior," but these are less common and typically require more advanced techniques to identify.
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Sound travels at an approximate speed of [tex]3.43(10^2)[/tex] m/s. How far will sound travel in 2 minutes?
Answer:41,160 meters in 2 minutes at the speed of 343 meters per second.
Step-by-step explanation:
The speed of sound varies depending on the medium it's traveling through, but assuming you meant the speed of sound in air at room temperature, it's approximately 343 meters per second.
To find out how far sound will travel in 2 minutes (120 seconds), we can simply multiply the speed of sound by the time:
Distance = Speed x Time
Distance = 343 m/s x 120 s
Distance = 41,160 meters
Therefore, sound will travel approximately 41,160 meters in 2 minutes at the speed of 343 meters per second.
Please upload a picture of a piece of paper with the problem worked out, and draw the graph for extra points, there will be 6 of these, so go to my profile and find the rest, and do the same, for extra points. for this one, use substitution method.
The value of X and y when substitution method is used to solve the given quadratic equation would be = 8 and 2 respectively.
How to calculate the unknown values using the substitution method?The equations that are given is listed below:
X - 3y = 2 ---> equation 1
2x - 6y = 6 ----> equation 2
In equation 1, make X the subject of formula;
X = 2 + 3y
Substitute X = 2 + 3y into equation 2,
2( 2 + 3y) - 6y = 6
4 + 6y - 6y = 6
y = 6-4
y = 2
Substitute y = 2 into equation 1;
x - 3(2) = 2
X = 2 + 6
X= 8
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Question 7 of 25
Emma choosing a weekly meeting time. She hopes to have two different
managers attend on a regular basis. The table shows the probabilities that
the managers can attend on the days she is proposing.
Monday
Wednesday
0. 82
0. 87
Manager A
Manager B
0. 88
0. 85
Assuming that manager A's availability is independent of manager B's
availability, which day should Emma choose to maximize the probability that
both managers will be available?
O A. Wednesday. The probability that both managers will be available is
0. 74
O B. Monday. The probability that both managers will be available is
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INTL
Emma should choose A. Wednesday to maximize the probability that both managers will be available.
A. Wednesday. The probability that both managers will be available is 0.74.
To calculate this, multiply the probabilities of each manager's availability for each day:
- Monday: Manager A (0.82) x Manager B (0.88) = 0.7216
- Wednesday: Manager A (0.87) x Manager B (0.85) = 0.7395
Since 0.7395 (Wednesday) is higher than 0.7216 (Monday), Emma should choose Wednesday to maximize the probability that both managers will be available.
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A particle, initially at rest, moves along the x-axis such that the acceleration at time t > 0 is given by a(t)= —sin(t) . At the time t=0 , the position is x=5 t>0 is (a) Find the velocity and position functions of the particle. b) For what values of time t is the particle at rest?
(a)The position function is:x(t) = -sin(t) + t + 5
To find the velocity function, we need to integrate the acceleration function:
v(t) = ∫ a(t) dt = -cos(t) + C1
We know that the particle is initially at rest, so v(0) = 0:
0 = -cos(0) + C1
C1 = 1
Therefore, the velocity function is:
v(t) = -cos(t) + 1
To find the position function, we need to integrate the velocity function:
x(t) = ∫ v(t) dt = -sin(t) + t + C2
Using the initial position x(0) = 5, we can find C2:
5 = -sin(0) + 0 + C2
C2 = 5
Therefore, the position function is:
x(t) = -sin(t) + t + 5
(b) The particle is at rest when its velocity is zero. So we need to solve for t when v(t) = 0:
0 = -cos(t) + 1
cos(t) = 1
t = 2πn, where n is an integer.
Therefore, the particle is at rest at times t = 2πn, where n is an integer.
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b. use the overhead rate in (a) to determine the amount of total and per-unit overhead allocated to each of the three products, rounded to the nearest dollar.
The amount of total and per-unit overhead allocated to each of the three products using overhead rate is equal to,
Total Per Unit Factory Overhead , Cost Factory Overhead Cost
Flutes $530 $1,060,000
Clarinets $795 $1,192,500
Oboes $397.5 $695,625
Total $1,722.5 $2,948,125
Budgeted factory overhead cost = $2,948,125
The single plantwide overhead rate
= Dividing the budgeted factory overhead cost by the total budgeted direct labor hours.
For this,
Flutes= 2,000×2
= 4,000 hours
Clarinets= 1,500×3
= 4,500 hours
Oboes= 1,750×1.5
= 2,625 hours
Total direct labor hours = 11,125
Substitute the value we have,
⇒ Single plantwide overhead rate = $2,948,125 / (2,000 x 2.0 + 1,500 x 3.0 + 1,750 x 1.5)
= $2,948,125 / 11,125
= $265 per direct labor hour
To allocate overhead to each product
=Multiply the overhead rate by the budgeted direct labor hours per unit for each product.
Substitute the value we have,
Flutes,
$265 x 2.0 = $530 total overhead cost, $265 per unit
Clarinets,
$265 x 3.0 = $795 total overhead cost, $265 per unit
Oboes,
$265 x 1.5 = $397.5 total overhead cost, $265 per unit
And
Total factory overhead cost allocated = Estimated manufacturing overhead rate× Actual amount of allocation base
For,
Flutes
= 4,000× 265
= $1,060,000
Clarinets
= 4,500×265
= $1,192,500
Oboes
= 2,625×265
= $695,625
This implies,
The total and per-unit overhead allocated to each product, rounded to the nearest dollar is,
Total Per Unit Factory Overhead , Cost Factory Overhead Cost
Flutes $530 $1,060,000
Clarinets $795 $1,192,500
Oboes $397.5 $695,625
Total $1,722.5 $2,948,125
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The above question is incomplete, the complete question is:
Bach Instruments Inc. makes three musical instruments: flutes, clarinets, and oboes. The budgeted factory overhead cost is $2,948,125. Overhead is allocated to the three products on the basis of direct labor hours. The products have the following budgeted production volume and direct labor hours per unit:
Budgeted Production Volume Direct Labor Hours Per Unit
Flutes 2,000 units 2.0
Clarinets 1,500 3.0
Oboes 1,750 1.5
a. Determine the single plantwide overhead rate.
$ per direct labor hour
b. Use the overhead rate in (a) to determine the amount of total and per-unit overhead allocated to each of the three products, rounded to the nearest dollar.
Total Per Unit
Factory Overhead Cost Factory Overhead Cost
Flutes $ $
Clarinets
Oboes
Total $
Look at picture please
Based on the inequality, 24.5x > 162 + 4.25x, Trina must sell more than 8 units of the handmade vases to make a profit.
What is inequality?Inequality refers to a mathematical statement that two or more algebraic expressions are unequal or inequivalent.
Mathematically, inequalities are depicted as:
Greater than (>)Greater than or equal to (≥)Less than (<)Less than or equal to (≤)Not equal to (≠).Selling price per handmade vase = $24.50
Variable cost per unit = $4.25
Fixed selling cost = $162
Let the number of vases to sell to make a profit = x
The total sales revenue = 24.5x
The total cost = 162 + 4.25x
To make a profit, 24.5x must be greater than 162 + 4.25x.
Inequality:24.5x > 162 + 4.25x
20.25x > 162
x > 8
Check:
Total sales revenue = $196 ($24.5(8)
Total cost = $196 ($162 + $4.25(8)
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Use the fundamental counting principle to find the total number of possible outcomes.
fitness tracker
battery 1 day, 3 days, 5 days, 7 days
color
silver, green, blue,
pink, black
Using the fundamental counting principle, there are 20 possible outcomes for the fitness tracker, considering its battery life and color options.
To find the total number of possible outcomes for the given criteria, we can use the fundamental counting principle. This principle states that if there are m ways to perform one task and n ways to perform another task, then there are m x n ways to perform both tasks together.
In this case, we have 4 options for battery life (1 day, 3 days, 5 days, and 7 days) and 4 options for color (silver, green, blue, pink, and black). Using the fundamental counting principle, we can multiply the number of options for battery life by the number of options for color to find the total number of possible outcomes:
4 options for battery life x 5 options for color = 20 possible outcomes.
Therefore, there are 20 possible combinations of battery life and color for a fitness tracker.
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the length of a retangle is 5 more than twoce its width its perimter is 88 feet find the dimensions use p=2l+2w
If length of rectangle is 5 more than twice it's width and having perimeter as 88 feet, then the dimensions of the rectangle are, length is 31 feet, width is 13 feet.
Let width of the rectangle be represented as = "w" feet.
It is given that, the length of rectangle is 5 more than twice it's width,
So, Length can be represented as "2w + 5" in feet;
We use formula for perimeter of rectangle, which is "P = 2Length + 2Width", where P = perimeter, L = length, and W = width.
In this case, we know that the perimeter is 88 feet, so we substitute the values,
We get,
⇒ 88 = 2(2w + 5) + 2w;
⇒ 88 = 4w + 10 + 2w,
⇒ 88 = 6w + 10,
⇒ 78 = 6w,
⇒ w = 13,
So the width is 13 feet. We use this value of width to find length of the rectangle:
⇒ L = 2w + 5,
⇒ L = 2(13) + 5,
⇒ L = 31
Therefore, the dimensions of the rectangle are 31 feet by 13 feet.
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Find the Lap lace transform of
f(t) = 6u (t- 2) + 3u(t-5) - 4u(t-6)
F(s)=
To find the Laplace transform of f(t), we use the formula:
L{f(t)} = ∫[0,∞) [tex]e^(-st)[/tex] f(t) dt
where L{f(t)} denotes the Laplace transform of f(t) and u(t) is the unit step function.
Using the linearity of the Laplace transform, we can find the Laplace transform of each term separately and add them up.
L{6u(t-2)} = [tex]6e^(-2s)[/tex] / s (applying the time-shift property)
L{3u(t-5)} = [tex]3e^(-5s)[/tex] / s (applying the time-shift property)
L{-4u(t-6)} = -[tex]4e^(-6s[/tex]) / s (applying the time-shift property)
Therefore, the Laplace transform of f(t) is:
F(s) = L{f(t)} = 6[tex]e^(-2s)[/tex] / s + [tex]3e^(-5s)[/tex] / s - [tex]4e^(-6s)[/tex]/ s
= [tex](6e^(-2s) + 3e^(-5s) - 4e^(-6s)) / s[/tex]
Hence, the Laplace transform of f(t) is F(s) = [tex](6e^(-2s) + 3e^(-5s) - 4e^(-6s)) / s.[/tex]
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Below is a table of hourly wages for receptionists working for various companies. What is the average wage for receptionists in this group? $9.67 $11.15 $11.60 $12.15 $14.50
The average wage for receptionists in this group is $11.60
How to calculate the average wageIt's important to note that "average wage" can be calculated in a few different ways, such as mean, median, and mode. Mean is the sum of all wages divided by the number of individuals, while median is the middle value in a range of wages. Depending on which method is used, the average wage figure can vary.
The table of hourly wages for receptionists working for various companies, the average wage for receptionists in this group will be:
= $58 / 5
= $11.60
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A rectangular brick wall is 6 wide and 1 m tall. Use Pythagoras' theorem to work out the distance between diagonally opposite corners. Give your answer in metres (m) to 1 d.p.
Pythagorean Theorem: a^2 + b^2 = c^2
---a and b are the legs of the triangle
---c is the hypotenuse/diagonal
a = 6
b = 1
c = ?
(6)^2 + (1)^2 = c^2
36 + 1 = c^2
37 = c^2
c = 6.0827
c (rounded) = 6.1
Answer = 6.1 meters
3
mpic
5
The ratio of union members to nonunion members working for a company is 4 to 5. If there are 100 union members working for the company, what is the total
number of employees?
Answer:
225
Step-by-step explanation:
since the ratio is 4 to 5 and the number of union workers are 100 you divide the number of union workers by their respective ratio which is four then multiply that by the 5
Maria is currently taking quantitative literacy course. The instructor often gives quizzes. Each quiz is worth
10 points. Maria got the following scores: 10, 9, 10, 9, 10.
(a) Calculate the average of her quizzes. Round your answer to the nearest tenth (if needed).
(b) Calculate standard deviation of her quizzes. Round your answer to the nearest tenth.
Maria's average quiz score is 9.6.
B. The standard deviation of her quizzes is approximately 0.5 (rounded to the nearest tenth).
What is the average?(a) The average of Maria's quizzes can be found by adding up all her scores and dividing by the total number of quizzes:
Average = (10 + 9 + 10 + 9 + 10) / 5 = 9.6
Therefore, Maria's average quiz score is 9.6.
(b) To calculate the standard deviation of her quizzes, we first need to find the variance. We can do this by finding the average of the squared differences between each score and the mean:
[(10 - 9.6)² + (9 - 9.6)² + (10 - 9.6)² + (9 - 9.6)² + (10 - 9.6)²] / 5 = 0.24
So the variance is 0.24. To find the standard deviation, we take the square root of the variance:
√0.24 ≈ 0.5
So the standard deviation of her quizzes is approximately 0.5 (rounded to the nearest tenth).
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A museum groundskeeper is creating a semicircular statuary garden with a diameter of 26 feet. There will be a fence around the garden. The fencing costs $9. 75 per linear foot. About how much will the fencing cost altogether? Round to the nearest hundredth. Use 3. 14 for π
The cost of fencing for the semicircular statuary garden is $651.99.
Finding the circumference of the full circle:
C = πd, where d is the diameter.
C = 3.14 × 26
C ≈ 81.64 feet
Since it's a semicircular garden, dividing the circumference by 2:
Semi-circular perimeter = 81.64 ÷ 2
Semi-circular perimeter ≈ 40.82 feet
Now, Adding the diameter to the semi-circular perimeter to get the total fence length:
Total fence length = 40.82 + 26
Total fence length ≈ 66.82 feet
Then, Calculating the total cost of fencing:
Cost = Total fence length × Cost per linear foot
Cost = 66.82 × $9.75
Cost ≈ $651.99
So, the cost of fencing the semicircular statuary garden will be approximately $651.99.
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The speed s in miles per hour that a car is traveling when it goes into a skid can be
estimated by the formula s = â 30fd, where f is the coefficient of friction and d is the length of the skid marks in feet. On the highway near Lake Tahoe, a police officer finds a car on the shoulder, abandoned by a driver after a skid and crash. He is sure that the driver was driving faster than the speed limit of 20 mi/h because the skid marks
measure 9 feet and the coefficient of friction under those conditions would be 0. 7. At about what speed was the driver driving at the time of the skid? Round your answer
to the nearest mi/h.
A. 23 mi/h
B. 189 mi/h
C. 14 mi/h
D. 19 mi/h
The driver was driving at a speed of about 14 mi/h at the time of the skid. option is C. 14 mi/h
Using the formula s = √(30fd), where f is the coefficient of friction (0.7) and d is the length of the skid marks in feet (9), we can estimate the speed at the time of the skid:
s = √(30 × 0.7 × 9)
s ≈ 14.53 mi/h
Rounding to the nearest mi/h, the driver was driving at approximately 15 mi/h at the time of the skid. However, none of the given options match this result. The closest option is C. 14 mi/h, so I would choose that as the best available answer.
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Which quadratic function represents the graph below?
the answer options are
y=3/14(x-5)(x+10)
y=3/14(x+5)(x-10)
y=1/3(x-5)(x+10)
y=1/3(x+5)(x-10)
y=3/14(x-5)(+10)
Step-by-step explanation:
.........................
For the function f(x,y) = x^2 e^{3xy}, find fx, and fy.
For function f(x,y) = x² e^{3xy} , fx = 2xe^{3xy} + 3x²y e^{3xy}, fy = 3x² e^{3xy}
The given function is f(x,y) = x² e^{3xy}.
To find the partial derivatives of f(x,y) with respect to x and y,
we differentiate the function with respect to each variable while treating the other variable as a constant.
To find fx, we differentiate the function f(x,y) with respect to x while treating y as a constant.
The derivative of x² is 2x, and the derivative of e^{3xy} is e^{3xy} times the derivative of 3xy with respect to x, which is 3y.
Therefore, we get:
fx = (d/dx)(x² e^{3xy}) = 2xe^{3xy} + 3x²y e^{3xy}
To find fy,
we differentiate the function f(x,y) with respect to y while treating x as a constant.
The derivative of e^{3xy} with respect to y is e^{3xy} times the derivative of 3xy with respect to y, which is 3x². Therefore, we get:
fy = (d/dy)(x² e^{3xy}) = 3x² e^{3xy}
Hence, the partial derivatives of f(x,y) are fx = 2xe^{3xy} + 3x^2y e^{3xy} and fy = 3x² e^{3xy}.
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Find the critical mumbers for g(x) = 2 sin r- r on (0,7). Then find the absolute maximum and minimum values for g(x) on (0,7). Give exact answers, not decimal approximations."
2 cos x - 1 = 0, cos x = 1/2, and x = π/3 or 5π/3. These are the critical numbers of g(x) on (0,7). the absolute maximum value of g(x) on (0,7) is √3 - π/3 and the absolute minimum value is -√3 - 5π/3.
To find the critical numbers for g(x) = 2sin(r) - r on the interval (0, 7), follow these steps:
1. Find the derivative of g(x): g'(x) = 2cos(r) - 1
2. Set the derivative equal to zero: 2cos(r) - 1 = 0
3. Solve for r: r = cos^(-1)(1/2)
Now, find the absolute maximum and minimum values for g(x) on the interval (0, 7):
1. Evaluate g(x) at the critical numbers: g(cos^(-1)(1/2)) = 2sin(cos^(-1)(1/2)) - cos^(-1)(1/2)
2. Evaluate g(x) at the endpoints of the interval: g(0) = 2sin(0) - 0, g(7) = 2sin(7) - 7
3. Compare the values from steps 1 and 2 to find the maximum and minimum values.
The critical numbers for g(x) = 2sin(r) - r on the interval (0, 7) are r = cos^(-1)(1/2). The absolute maximum and minimum values for g(x) on the interval (0, 7) can be found by comparing g(cos^(-1)(1/2)), g(0), and g(7).
To find the critical numbers of g(x) = 2 sin x - x on (0,7), we first need to find its derivative:
g'(x) = 2 cos x - 1
Setting g'(x) = 0, we get:
2 cos x - 1 = 0
cos x = 1/2
x = π/3 or 5π/3
These are the critical numbers of g(x) on (0,7).
To find the absolute maximum and minimum values of g(x) on (0,7), we need to evaluate g(x) at the critical numbers and at the endpoints of the interval (0,7).
g(0) = 0
g(π/3) = 2 sin(π/3) - π/3 = √3 - π/3
g(5π/3) = 2 sin(5π/3) - 5π/3 = -√3 - 5π/3
g(7) = 2 sin(7) - 7
To determine the absolute maximum and minimum values, we compare these values:
The absolute maximum value is √3 - π/3, which occurs at x = π/3.
The absolute minimum value is -√3 - 5π/3, which occurs at x = 5π/3.
Therefore, the absolute maximum value of g(x) on (0,7) is √3 - π/3 and the absolute minimum value is -√3 - 5π/3.
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a time capsule has been buried 98m away from the cave at a bearing of 312 degrees how far west of the cave is the time capsule buried? give your answer in 1 decimal places
If a time capsule has been buried 98m away from the cave at a bearing of 312. the time capsule is buried about 82.2 meters west of the cave.
What is the time capsule?To find how far west the time capsule is buried, we need to find the horizontal component of the displacement vector that points from the cave to the location of the time capsule. We can use trigonometry to do this:
cos(312°) = adjacent/hypotenuse
The hypotenuse is the distance between the cave and the time capsule, which is 98m. The adjacent side represents the horizontal distance between the two points, which is what we want to find. Rearranging the equation, we get:
adjacent = cos(312°) x hypotenuse
adjacent = cos(312°) x 98
adjacent ≈ 82.2
Therefore, the time capsule is buried about 82.2 meters west of the cave.
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A box contains green marbles and blue marbles. Yosef shakes the box and chooses a marble at random. He records the color, then places the marble back into the box. Yosef repeats the process until he chooses 50 marbles. The table shows the count for each color. Write a probability model for choosing a marble.
green: 36
blue: 14
The probability model for choosing a marble from the box is P(green) = 36/50 and P(blue) = 14/50.
To create this probability model, first, count the total number of marbles chosen, which is 50. Then, count the number of green and blue marbles chosen, which are 36 and 14, respectively.
Divide the number of each color by the total number of marbles to find the probability of choosing a green or blue marble.
P(green) is calculated as 36/50 or 0.72, and P(blue) is calculated as 14/50 or 0.28. This model represents the likelihood of choosing a green or blue marble from the box based on Yosef's experiment.
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Chris is using the expression 4x + 2 to represent the number of students in his gym class. There are four times as many students as basketballs, and there are two more students in the locker room. What does x represent? (4 points)
X represents the number of basketballs in Chris's gym class.
Chris represents the number of students in his gym class with the expression 4x + 2. We know that the number of students is four times the number of basketballs, so we can set up the equation 4x = the number of basketballs.
If we substitute this expression for the number of students in the gym class, we get 4(4x) + 2 = the total number of students in the gym class and locker room. We also know that there are two more students in the locker room, so we can add 2 to this expression to get 4(4x) + 4 = the total number of students in the gym class and locker room.
Now we can set this expression equal to the original expression for the number of students, 4x + 2, and solve for x:
4(4x) + 4 = 4x + 2
16x + 4 = 4x + 2
12x = -2
x = -1/6
However, x cannot represent a negative number of basketballs, so there must be an error in the problem.
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Which cardboard box can hold the greatest number of 1 in x 2 in x 4 in sponges
The cardboard box with the largest volume can hold the greatest number of 1 in x 2 in x 4 in sponges.
To find the box with the largest volume, first determine the volume of each sponge: V_sponge = 1 in x 2 in x 4 in = 8 cubic inches. Next, find the volume of each box by multiplying its length, width, and height (V_box = L x W x H).
To determine how many sponges each box can hold, divide the volume of the box by the volume of the sponge (V_box / V_sponge). The box with the highest resulting quotient can hold the most 1 in x 2 in x 4 in sponges.
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Aria ate the pictured slice of pizza. If the original
pizza was 8 inches in diameter, what is the area
of the slice she ate?
Answer:
Step-by-step explanation:
50.24n2
Find the area of the shaded region. Round your answer to the nearest hundredth.
Answer:
The radius of the circle is 5/√2 = (5√2)/2 inches.
Area of circle = π((5√2)/2)^2
= 25π/2 square inches
Area of triangle = (1/2)(5√2)((5√2)/2)
= 25/2 square inches
Area of shaded region
= (25/2)(π - 1) = 26.77 square inches
PLEASE HELPPPP!! 20pts Students in the Drama Club are purchasing accessories for a play. They shop at two different stores over the span of three days. The items purchased at Store A al cost the same amount. The tems pur
⢠Day 1: Students spent $30. They purchased 4 items from Store A and 7 items from Store B.
⢠Day 2: Students spent $22. They purchased 3 items from Store A and 5 items from Store B.
On Day 3 students will need to buy 10 items from Store A and 17 items from Store B. What is the amount of money the students will need on the third day?
Part A: Write a system of equations to model the situations
The students will need $74 on the third day.
Let x be the cost of one item at Store A and y be the cost of one item at Store B. Then the system of equations to model the situation is:
4x + 7y = 30
3x + 5y = 22
To find the cost on Day 3, we need to solve for x and y, and then use those values to calculate:
10x + 17y = ?
Part B: Solve the system of equations to find x and y
To solve the system of equations, we can use elimination or substitution. Here, we'll use substitution.
From the first equation, we can solve for x:
4x + 7y = 30
4x = 30 - 7y
x = (30 - 7y)/4
Substitute this expression for x into the second equation:
3x + 5y = 22
3((30 - 7y)/4) + 5y = 22
(90 - 21y)/4 + 5y = 22
90 - 21y + 20y = 88
-y = -2
y = 2
Now that we know y = 2, we can substitute this value back into either equation to find x:
4x + 7y = 30
4x + 7(2) = 30
4x + 14 = 30
4x = 16
x = 4
So x = 4 and y = 2.
Part C: Calculate the amount of money needed on Day 3
Finally, we can use these values to calculate the amount of money needed on Day 3:
10x + 17y = 10(4) + 17(2) = 40 + 34 = 74
Therefore, the students will need $74 on the third day.
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Question 1(Multiple Choice Worth 2 points)
(Circle Graphs MC)
The circle graph describes the distribution of preferred transportation methods from a sample of 400 randomly selected San Francisco residents.
circle graph titled San Francisco Residents' Transportation with five sections labeled walk 40 percent, bicycle 8 percent, streetcar 15 percent, bus 10 percent, and cable car 27 percent
Which of the following conclusions can we draw from the circle graph?
Together, Streetcar and Cable Car are the preferred transportation for 168 residents.
Together, Walk and Streetcar are the preferred transportation for 55 residents.
Bus is the preferred transportation for 45 residents.
Bicycle is the preferred transportation for 50 residents.
Question 2(Multiple Choice Worth 2 points)
(Appropriate Measures MC)
The box plot represents the number of tickets sold for a school dance.
A horizontal line labeled Number of Tickets sold that starts at 8, with tick marks every one unit up to 30. The graph is titled Tickets Sold for A Dance. The box extends from 17 to 21 on the number line. A line in the box is at 19. The lines outside the box end at 10 and 27.
Which of the following is the appropriate measure of center for the data, and what is its value?
The mean is the best measure of center, and it equals 19.
The median is the best measure of center, and it equals 4.
The median is the best measure of center, and it equals 19.
The mean is the best measure of center, and it equals 4.
Question 3(Multiple Choice Worth 2 points)
(Comparing Data LC)
The histograms display the frequency of temperatures in two different locations in a 30-day period.
A graph with the x-axis labeled Temperature in Degrees, with intervals 60 to 69, 70 to 79, 80 to 89, 90 to 99, 100 to 109, 110 to 119. The y-axis is labeled Frequency and begins at 0 with tick marks every one unit up to 14. A shaded bar stops at 10 above 60 to 69, at 9 above 70 to 79, at 5 above 80 to 89, at 4 above 90 to 99, and at 2 above 100 to 109. There is no shaded bar above 110 to 119. The graph is titled Temps in Sunny Town.
A graph with the x-axis labeled Temperature in Degrees, with intervals 60 to 69, 70 to 79, 80 to 89, 90 to 99, 100 to 109, 110 to 119. The y-axis is labeled Frequency and begins at 0 with tick marks every one unit up to 16. A shaded bar stops at 2 above 60 to 69, at 4 above 70 to 79, at 12 above 80 to 89, at 6 above 90 to 99, at 4 above 100 to 109, and at 2 above 110 to 119. The graph is titled Temps in Desert Landing.
When comparing the data, which measure of center should be used to determine which location typically has the cooler temperature?
Median, because Desert Landing is symmetric
Mean, because Sunny Town is skewed
Mean, because Desert Landing is symmetric
Median, because Sunny Town is skewed
Question 4(Multiple Choice Worth 2 points)
(Appropriate Measures MC)
A charity needs to report its typical donations received. The following is a list of the donations from one week. A histogram is provided to display the data.
10, 11, 35, 39, 40, 42, 42, 45, 49, 49, 51, 51, 52, 53, 53, 54, 56, 59
A graph titled Donations to Charity in Dollars. The x-axis is labeled 10 to 19, 20 to 29, 30 to 39, 40 to 49, and 50 to 59. The y-axis is labeled Frequency. There is a shaded bar up to 2 above 10 to 19, up to 2 above 30 to 39, up to 6 above 40 to 49, and up to 8 above 50 to 59. There is no shaded bar above 20 to 29.
Which measure of variability should the charity use to accurately represent the data? Explain your answer.
The range of 13 is the most accurate to use, since the data is skewed.
The IQR of 49 is the most accurate to use to show that they need more money.
The range of 49 is the most accurate to use to show that they have plenty of money.
The IQR of 13 is the most accurate to use, since the data is skewed.
Question 5(Multiple Choice Worth 2 points)
(Making Predictions MC)
A recent conference had 900 people in attendance. In one exhibit room of 80 people, there were 65 teachers and 15 principals. What prediction can you make about the number of principals in attendance at the conference?
There were about 820 principals in attendance.
There were about 731 principals in attendance.
There were about 208 principals in attendance.
There were about 169 principals in attendance.
Question 6(Multiple Choice Worth 2 points)
(Creating Graphical Representations LC)
A teacher was interested in the subject that students preferred in a particular school. He gathered data from a random sample of 100 students in the school and wanted to create an appropriate graphical representation for the data.
Which graphical representation would be best for his data?
Stem-and-leaf plot
Histogram
Circle graph
Box plot
Answer 1: Together, Streetcar and Cable Car are the preferred transportation for 168 residents.
How to solveThe circle graph shows the percentage of residents who prefer each transportation method, and the total sample size is 400.
For streetcar, (15/100) x 400 = 60 residents prefer it, and for cable car, (27/100) x 400 = 108 residents prefer it.
Together, Streetcar and Cable Car are the preferred transportation for 60 + 108 = 168 residents.
Answer 2: The median is the best measure of center, and it equals 19.
The box plot shows the distribution of the number of tickets sold for a school dance.
The median is the middle value of the data when arranged in order, and it is represented by the line in the box. In this case, the median is 19. The mean, on the other hand, can be influenced by extreme values, and we cannot determine it from the box plot alone.
Answer 3: Median, because Sunny Town is skewed.
When comparing the data, we need to consider the measure of center that is less affected by extreme values, and that is the median.
The median is the middle value of the data when arranged in order. The histogram for Sunny Town is skewed to the right, which means that there are some very high values that are affecting the mean.
Therefore, the median is the better measure of center to determine which location typically has the cooler temperature.
Answer 4: The IQR of 13 is the most accurate to use, since the data is skewed.
The histogram shows the frequency of donations received by a charity, and the data is skewed to the right.
The IQR (Interquartile Range) is the difference between the third quartile (Q3) and the first quartile (Q1), which represents the middle 50% of the data.
The IQR is less sensitive to extreme values and is a better measure of variability for skewed data. In this case, the IQR is 49 - 42 = 7, which is the most accurate measure of variability to use.
Answer 5: There were about 15 principals in attendance.
In the exhibit room, out of 80 people, 15 are principals.
We can assume that the proportion of principals in the exhibit room is the same as the proportion of principals in the conference.
Therefore, the estimated number of principals in the conference is (15/80) x 900 = 168.75, which is approximately 169.
Answer 6: Histogram
The teacher wants to represent the subject preferences of 100 students. A histogram would be the best graphical representation to use because it shows the frequency distribution of a continuous variable, which in this case could be the number of students who prefer each subject.
A stem-and-leaf plot is used for small datasets, and a box plot is used to display the distribution of a continuous variable across categories. A circle graph is more appropriate for displaying categorical data, such as the percentage of students who prefer each subject.
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The radius of a circle is increasing uniformly at the rate of 5cm/sec. Find the rate at which the area of the circle is increasing when the radius is 6 cm.
When the radius is 6cm, the rate at which the area of the circle is increasing is 60π cm^2/sec.
To find the rate at which the area of the circle is increasing, we need to use the formula for the area of a circle: A = πr^2. We can differentiate both sides of this equation with respect to time to get:
dA/dt = 2πr(dr/dt)
where dA/dt is the rate at which the area of the circle is increasing, dr/dt is the rate at which the radius is increasing (which we know is 5cm/sec), and r is the current radius of the circle.
So, when the radius is 6cm, we have:
r = 6cm
dr/dt = 5cm/sec
Plugging these values into the formula above, we get:
dA/dt = 2π(6cm)(5cm/sec)
dA/dt = 60π cm^2/sec
Therefore, when the radius is 6cm, the rate at which the area of the circle is increasing is 60π cm^2/sec.
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Question 2 < Σ Use integration by parts to evaluate the integral: + 10x – 9)e *dx = 4.1 f(x²
The value of integral ∫(x^2+10x-9)e^-4x dx is (-1/4)(x^2+10x-9)e^-4x - (1/8)(x+5)e^-4x - (1/32)e^-4x + C.
To evaluate the integral ∫(x^2+10x-9)e^-4x dx using integration by parts, we need to choose two functions to multiply together, one of which we differentiate and the other we integrate. A common choice is to let u = x^2+10x-9 and dv = e^-4x dx, which gives du = (2x+10) dx and v = (-1/4)e^-4x.
Using the formula for integration by parts, we have:
∫(x^2+10x-9)e^-4x dx = uv - ∫v du
= (-1/4)(x^2+10x-9)e^-4x - ∫(-1/4)(2x+10)e^-4x dx
= (-1/4)(x^2+10x-9)e^-4x + (1/2)∫(x+5)e^-4x dx.
Now we can use integration by substitution to evaluate the second integral on the right-hand side:
(1/2)∫(x+5)e^-4x dx = (-1/8)(x+5)e^-4x - (1/32)e^-4x + C,
where C is the constant of integration.
Substituting this back into our previous equation, we get:
∫(x^2+10x-9)e^-4x dx = (-1/4)(x^2+10x-9)e^-4x - (1/8)(x+5)e^-4x - (1/32)e^-4x + C.
Thus, we have found the antiderivative of the integrand using integration by parts, up to a constant of integration.
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Complete question is:
Use integration by parts to evaluate the integral: ∫(x^2+ 10x – 9)e ^-4x dx
Jane moved from a house with 78 square feet of closet space to an apartment with 47.58 square feet of closet space. what is the percentage decrease of jane's closet space?
Jane's closet space decreased by approximately 38.97%.
To find the percentage decrease of Jane's closet space, we need to first calculate the amount of decrease and then express it as a percentage of the original value.
The amount of decrease is the difference between the original closet space and the new closet space:
Decrease = Original closet space - New closet space
Decrease = 78 - 47.58
Decrease = 30.42
So Jane's closet space decreased by 30.42 square feet.
To express this decrease as a percentage of the original value, we use the following formula:
Percentage decrease = (Decrease / Original value) x 100%
Substituting the values, we get:
Percentage decrease = (30.42 / 78) x 100%
Percentage decrease ≈ 38.97%
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