Find the center and radius of circle x^2 + y^2 -12x + 2y + 33 = 0
Answer:
To find the center and radius of the circle, we need to rewrite the equation in standard form, which is: (x - h)^2 + (y - k)^2 = r^2 where (h, k) is the center of the circle and r is the radius. To rewrite the given equation in standard form, we need to complete the square for both x and y terms. Let's start with the x terms: x^2 - 12x = -(y^2 - 2y + 33) To complete the square for x, we need to add and subtract (12/2)^2 = 36: x^2 - 12x + 36 - 36 = -(y^2 - 2y + 33) (x - 6)^2 - 36 = -(y^2 - 2y + 33) Now
To find the center and radius of the circle given by the equation:
x^2 + y^2 -12x + 2y + 33 = 0
We need to rewrite the equation in standard form, which is:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
To do this, we need to complete the square for both x and y terms by adding and subtracting appropriate terms inside the parentheses.
(x^2 - 12x) + (y^2 + 2y) + 33 = 0
To complete the square for x terms, we need to add (12/2)^2 = 36 inside the first set of parentheses and subtract it from the equation:
(x^2 - 12x + 36) + (y^2 + 2y) + 33 - 36 = 0
(x - 6)^2 + (y^2 + 2y) - 3 = 0
To complete the square for y terms, we need to add (2/2)^2 = 1 inside the second set of parentheses and subtract it from the equation:
(x - 6)^2 + (y^2 + 2y + 1) - 4 = 0
(x - 6)^2 + (y + 1)^2 = 4
Now we can see that the equation is in standard form, where (h, k) = (6, -1) is the center of the circle, and r^2 = 4, so the radius is r = 2.
Therefore, the center of the circle is (6, -1) and the radius is 2.
If 75% of Garden A’s data is 12.75 and 50% of Garden B’s data is 15 than how much more percent is Garden B than A.
The Garden B has 76.47% more data than Garden A.
What is percentage?Percentage is a way of expressing a number as a fraction of 100. It is a commonly used method to represent proportions, rates, and percentages in various fields such as finance, economics, and statistics.
Let's first find the actual values of Garden A's and Garden B's data:
For Garden A, we know that 75% of its data is equal to 12.75. We can use this information to set up an equation:
0.75x = 12.75
Solving for x, we get:
x = 12.75 / 0.75 = 17
Therefore, Garden A's total data is 17.
For Garden B, we know that 50% of its data is equal to 15. We can use this information to set up an equation:
0.5x = 15
Solving for x, we get:
x = 15 / 0.5 = 30
Therefore, Garden B's total data is 30.
Now, to find how much more percent Garden B has than Garden A, we can use the formula:
percent difference = |(Garden B - Garden A) / Garden A| * 100
Plugging in the values, we get:
percent difference = |(30 - 17) / 17| * 100 = 76.47%
Therefore, Garden B has 76.47% more data than Garden A.
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In the diagram, PC // DF, AB // DE, EDF = 84° QBD = 74° and PCD = 148°. Calculate (a) ABQ (b) CDE
Using the alternate angles theorem,
(a) The measure of angle ABQ is 54°
(b) The measure of angle CDE is 128°
Calculating the measure of anglesFrom the question, we are to determine the measure of the unknown angles
First, let us produce line BCD to Z
Then,
m ∠FDZ = m ∠BCP
From the given diagram
m ∠BCP + 148° = 180° (Sum of angles on a straight line)
m ∠BCP = 180° - 148°
m ∠BCP = 32°
Therefore,
m ∠FDZ = 32°
Also,
m ∠EDZ = 84° - m ∠FDZ
m ∠EDZ = 84° - 32°
m ∠EDZ = 52°
m ∠CDE + m ∠EDZ = 180° (Sum of angles on a straight line)
m ∠CDE + 52° = 180°
m ∠CDE = 180° - 52°
m ∠CDE = 128°
m ∠ABQ + m ∠QBD = m ∠CDE (Alternate interior angles)
m ∠ABQ + 74° = 128°
m ∠ABQ = 128° - 74°
m ∠ABQ = 54°
Hence,
m ∠ABQ is 54°
and
m ∠CDE is 128°
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To build your own capacitor, you have cut your sheets of wax paper and aluminum foil to a length of L = 0.8 m and you are rolling them on a pipe of diameter D = 5 cm. What is the number of turns? -n = 2.6 -n = 20.4 -n = 5.1 -n=10.2
The number of turns in the capacitor is approximately 5.1.
How we get the number of turns (n)?The number of turns in the capacitor can be calculated using the formula:
n = L / [π(D + t)]where L is the length of the sheets, D is the diameter of the pipe, and t is the thickness of the sheets.
Using the provided values, we can calculate the number of turns as follows:
L = 0.8 mD = 5 cm = 0.05 mt = negligible (assumed to be much smaller than D)n = 0.8 / [π(0.05)]= 5.092 (rounded to one decimal place)
In this problem, we are asked to calculate the number of turns in a capacitor made from sheets of wax paper and aluminum foil that are rolled on a pipe of diameter 5 cm. We are given the length of the sheets as 0.8 m.
To calculate the number of turns, we can use the formula that relates the length of the sheets, diameter of the pipe, and thickness of the sheets to the number of turns in the capacitor. Assuming the thickness of the sheets is much smaller than the diameter of the pipe, we can ignore the thickness in the calculation.
Using the provided values and this formula, we find that the number of turns is approximately 5.1. This means that the sheets need to be rolled around the pipe about 5 times to create the capacitor.
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please help with this geometry hmw
For two triangles to be similar by side-angle-side (SAS) similarity, they must have two pairs of proportional sides and an included angle that is congruent between them.
How to prove similarity?
Let's consider triangles FGH and JKL. By inspection, we can see that FG is proportional to JK and GH is proportional to KL. Additionally, the included angles ∠FGH and ∠JKL are congruent, which means that we have two pairs of proportional sides and an included congruent angle.
Therefore, we can conclude that triangles FGH and JKL are similar by SAS similarity. This means that the corresponding angles of the two triangles will be congruent, and the corresponding sides will be in proportion to each other.
We can use this similarity to find missing side lengths or angles. For example, if we know the length of FG in triangle FGH and the length of JK in triangle JKL, we can use their proportionality to find the length of GH or KL. Similarly, we can use the congruent angles to find other missing angles in either triangle.
It is important to note that if we were to show that two triangles are similar by SAS, we must show that no other pair of sides or angles are congruent between the two triangles. If there were another pair of congruent sides or angles, we would need to use a different similarity criterion such as angle-angle (AA) or side-side-side (SSS).
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(Using trig to find an angle)
Solve for x. Round to the nearest tenth of a degree, if necessary.
Step-by-step explanation:
For a RIGHT triangle Sin = opposite LEG / Hypotenuse
sin x = 45/ 83
x = arcsin (45/83) = use calculator = 32.8 degrees
Please help I’ve posted this question twice and it’s due tonight
Based on the Venn diagram, the results in order are:
Students who do not like DC superheroes: 117Students who like Marvel superheroes: 358Students who like both DC and Marvel superheroes: 155Students who like DC but not Marvel superheroes: 85Students who like Marvel but not DC superheroes: 203Students who do not like the DC or Marvel superheroes: 117How do we solve this?To solve this problem, we can use a Venn diagram to represent the information given.
Total students surveyed = 560
395 like superheroes from either company
240 prefer DC superheroes
358 do not like Marvel superheroes
To find how many students did not like DC superheroes, we need to find the number of students who do not fall into the DC circle:
Students who do not like Marvel superheroes = 358
Students who like both DC and Marvel superheroes = (395 - 240) = 155
Students who like Marvel superheroes only = ?
Students who like DC superheroes only = (240 - 155) = 85
To find how many students like Marvel superheroes, we need to find the number of students who fall into the Marvel circle:
Students who do not like DC superheroes = ?
Students who like both DC and Marvel superheroes = 155
Students who like Marvel superheroes only = ?
Students who like DC superheroes only = 85
To find how many students like both DC and Marvel superheroes, we can use the information we already have:
Students who like both DC and Marvel superheroes = 155
To find how many students like DC but not Marvel superheroes, we can use the information we already have:
Students who like DC superheroes only = 85
To find how many students like Marvel but not DC superheroes, we can use the information we already have:
Students who like Marvel superheroes only = (358 - 155) = 203
Finally, to find how many students do not like the DC or Marvel superheroes, we need to find the number of students who do not fall into either circle:
Students who do not like DC superheroes = ?
Students who like both DC and Marvel superheroes = 155
Students who like Marvel superheroes only = 203
Students who like DC superheroes only = 85
We can now use the formula for the total number of students surveyed to find the number of students who do not like DC or Marvel superheroes:
Total students surveyed = Students who do not like DC superheroes + Students who like both DC and Marvel superheroes + Students who like Marvel superheroes only + Students who like DC superheroes only
560 = Students who do not like DC superheroes + 155 + 203 + 85
Students who do not like DC superheroes = 117
Similarly, we can use the same formula to find the number of students who like Marvel superheroes:
560 = Students who do not like DC superheroes + 155 + Students who like Marvel superheroes only + 85
Students who like Marvel superheroes = 203 + 155
Students who like Marvel superheroes = 358
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Faith kicks a football. The graph below shows the height of the football ℎ h in feet after � t seconds.
Answer:
faith
Step-by-step explanation:
you are a Beautiful human being
Emilio flipped a pair of fair coins 60 times. On 24 occasions, heads appeared on both coins. How is this outcome different from the expected outcome?
Emiliο's οutcοme οf getting heads οn bοth cοins 24 times in 60 flips is 9 mοre than the expected οutcοme.
What is prοbability?Prοbability is simply hοw likely sοmething is tο happen.
The expected οutcοme fοr flipping a pair οf fair cοins οnce is that there is a 1/4 (οr 0.25) prοbability οf getting heads οn bοth cοins.
Therefοre, the expected number οf times that heads wοuld appear οn bοth cοins in 60 flips is:
Expected number = (Prοbability οf heads οn bοth cοins) x (Number οf flips)
Expected number = 0.25 x 60
Expected number = 15
Emiliο οbserved that heads appeared οn bοth cοins 24 times, which is greater than the expected number οf 15.
The difference between the οbserved οutcοme and the expected οutcοme is:
Difference = Observed number - Expected number
Difference = 24 - 15
Difference = 9
Therefοre, Emiliο's οutcοme οf getting heads οn bοth cοins 24 times in 60 flips is 9 mοre than the expected οutcοme.
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a scientist claims that 6% of viruses are airborne. if the scientist is accurate, what is the probability that the proportion of airborne viruses in a sample of 403 viruses would differ from the population proportion by greater than 3% ? round your answer to four decimal places.
The probability that the proportion of airborne virus in the sample of 553 viruses is greater than 5% is 0.1151.
According to the central limit theorem, if a large sample of n > 30 is selected from an unknown population and the sample proportion of each sample is calculated, the sampling distribution of the sample proportion obeys the normal distribution.
The mean of the sampling distribution (U) for this sample proportion is:
U = p
The standard deviation (p) of the sampling distribution for this sample proportion is:
U = √ ((p(1 - p)) /n )
The sample size is, n = 553 > 30 so the central limit theorem applies in this case.
Up = √((p(1 - p))/n) = √((0.04 * 0.96)/553) = 0.0083
Calculate the probability that the proportion of virus in the air in 553 virus samples is greater than 3% as follows:
P(p >0.03)
= P (p - Up > (0.
= 0.11507
≈ ≈ 0.1151
Therefore, the probability that the proportion of airborne virus in a sample of 553 viruses is greater than 3% is 0.1151.
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Which equation best matches the graph show below?
This equation is a parabola with a vertex located at (-1, -2). The equation that best matches the graph shown below is y= -5(x+1)²-2.
What is parabola?It is the graph of a quadratic function and forms a symmetrical U-shape.
The graph shows that the parabola is opening downward and has an x-intercept at (1, 0). By examining the graph and the equations, it can be seen that the equation y= -5(x+1)²-2 is the only equation that describes the graph.
The equation y= -5(x+1)²-2 can be rearranged to the form y= a(x-h)²+k. This can be seen by expanding the equation to y= -5x²-10x-2.
The a, h, and k values can be calculated as follows:
a = -5, h = -1, and k = -2.
This equation is an example of a quadratic equation in the form y = a(x-h)²+k, where a is the coefficient of the x squared term, h is the x-value of the vertex, and k is the y-value of the vertex.
The graph shown is a parabola with a vertex located at (-1, -2), an x-intercept at (1, 0), and an equation of y= -5(x+1)²-2.
These characteristics are all accurately represented by the equation
y= -5(x+1)²-2. This equation is the only equation that matches the graph.
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Use the Triangle Inequality Theorem to name a segment for the third side of a triangle if no segments are congruent and the first two sides are AB and BD. Use this key to enter the response: AB=1, AC=2, AD=3, BC=5, BD=6, CD=7. List them in ascending order. (fill in the blank)
__or__
(1 point)
The lengths of the sides in ascending order are AB=1, BD=6, CD=7.
What is length?Length is a measurement of how long an object is in one dimension, usually referring to its distance from one end to the other. It is a fundamental physical quantity and is often measured in units such as meters, centimeters, feet, or inches, depending on the context and system of measurement being used. Length can be used to describe the size of physical objects, such as the length of a piece of wood or the distance between two points in space, as well as more abstract concepts, such as the length of time that has passed or the length of a written document.
According to the Triangle Inequality Theorem, the length of any side of a triangle must be less than the sum of the lengths of the other two sides.
If AB=1 and BD=6, then the third side must be less than AB+BD=1+6=7. Therefore, CD<7.
If AC=2 and BD=6, then the third side must be less than AC+BD=2+6=8. Therefore, CD<8.
If AD=3 and BD=6, then the third side must be less than AD+BD=3+6=9. Therefore, CD<9.
Since CD is less than all of these values, the possible length for CD is the smallest value, which is CD=7. Therefore, the segment for the third side of the triangle is CD.
The lengths of the sides in ascending order are AB=1, BD=6, CD=7.
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is there a realationship between a students gpa and the bumvber of pencils in his or her backpack? jordynn and angie decided to find out by selecting a random sample of students from their high school
It is unlikely that carrying more or fewer pencils in one's backpack would have a direct impact on GPA.
How to find relationship ?To determine if there is a relationship between a student's GPA and the number of pencils in their backpack, Jordynn and Angie would need to conduct the following steps:
If a correlation is found, then it suggests that there may be a relationship between the students' GPA and the number of pencils in their backpack. However, in this case, it is unlikely that carrying more or fewer pencils in one's backpack would have a direct impact on GPA.
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i need help badly because im confused
Answer:
92in²
Step-by-step explanation:
A= 1/2(b1+b2)h = 1/2 (18+5)8=
1/2 (23)8= 1/2 184 = 92in²
Find the volume of the composite figure.
The volume of the triangular prism is given,
find the volume of the rectangular prism.
Triangular Prism
Volume = 126 cm³
Rectangular Prism
Volume = [?]cm³
7 cm
8 cm
6 cm
6 cm
Hint: V w.h
Please help 30 points
The volume of the rectangular prism is 336 [tex]cm^{3}[/tex] and the volume of the composite figure is 462 [tex]cm^{3}[/tex].
How to find the volume of the composite figure?Measure the lowest solid figure's dimensions and calculate its volume.
Calculate the volume by taking measurements of the upper solid figure.
Adding the volumes of the two solid figures will allow you to calculate the volume of the combined figure.
The volume of rectangular prism v = l * w * h
= 8 * 7 * 6
= 336 [tex]cm^{3}[/tex]
The volume of the triangular prism = 126 [tex]cm^{3}[/tex]
The volume of the composite figure = 336 [tex]cm^{3}[/tex] + 126 [tex]cm^{3}[/tex]= 462 [tex]cm^{3}[/tex]
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Two pools are being filled with water. To start, the first pool contains 1900 liters of water and the second pool contains 1501 liters of water. Water is being added to the first pool at a rate of 21.25 liters per minute. Water is being added to the second pool at a rate of 35.5 liters per minute.
To find the amount of water in each pool at any given time, we need to use the formula:
Amount of water = Initial amount of water + (rate of inflow × time)
Let's say we want to find the amount of water in each pool after 10 minutes. Using the formula, we get:
Amount of water in first pool = 1900 + (21.25 × 10) = 2125 liters
Amount of water in second pool = 1501 + (35.5 × 10) = 501 liters
Therefore, after 10 minutes, the first pool contains 2125 liters of water and the second pool contains 501 liters of water.
PLEASE HELP URGENT. I WILL GIVE OUT MANY POINTS!! WILL MARK BRANLIEST RIGHT ANSWER.
Please note: when it says △ABH it means the angles not the sides. there are also sides called a b and h. the angles are capitalized and the sides are lowercase
Use triangle ABH to explain why sin A=cos(90°-A)
Answer:
sin A = cos B = a/h
A and B are complements of each other, so A + B = 90°. It follows that B = 90° - A, meaning sin A = cos(90° - A).
Find the angle measure indicated. Assume that lines which appear tangent are tangent. A. 110 degrees B. 175 degrees C. 155 degrees D. 120 degrees
Therefore, the angle measure indicated 110° that is option A.
What is angle?An angle is the measure of the amount of rotation between two intersecting lines or planes. It is defined as the space between two intersecting lines or planes measured in degrees, radians, or other units of angular measurement. Angles are often represented by a symbol that looks like a small arc, with three letters labeling the points that define the angle, such as "ABC." The vertex of an angle is the point where the two lines or planes intersect, and the sides of an angle are the two rays that extend from the vertex. Angles are used in geometry, trigonometry, and other mathematical fields to measure and analyze the relationships between shapes and objects.
Here,
Since sector PF has a measure of 90 degrees, then angle PAF must also have a measure of 90 degrees since it is an inscribed angle that intercepts the same arc.
To find the angle measure of QSR, we first need to find the measure of angle QPR.
Since sector PF is a 90 degree sector and line QR is tangent to the circle at point R, we know that angle QRP is a right angle (90 degrees).
Similarly, since sector QR is a 150 degree sector and line PS is tangent to the circle at point S, we know that angle QPS is a right angle (90 degrees).
Therefore, angle QPR + angle QPS = 90 + 90 = 180 degrees.
Since angle QPS and angle QRP share ray QR, they are adjacent angles.
So, angle QSR = 360 - (angle QPR + angle QPS)
= 360 - 180 -70
= 110 degrees
Therefore, the measure of angle QSR is 110 degrees, which is equivalent to a straight line.
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At arcade palace, a child can buy 16 video game tokens for $4. At Alibaba's,
At Arcade Palace, each video game token costs 0.25.
At Arcade Palace, a child can buy 16 video game tokens for 4.
The total number or cost of something is the number or cost that you get when you add together or count all the parts
in it.
A number is a mathematical value used for counting or measuring or labeling objects. Numbers are used to performing
arithmetic calculations.
To find the cost per token, follow these steps:
Determine the total number of tokens (16) and the total cost (4).
Divide the total cost by the total number of tokens to find the cost per token.
In this case, 4 ÷ 16 tokens = 0.25 per token.
Therefore, at Arcade Palace, each video game token costs 0.25.
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Determine if 0.2 (18y) is equivalent to 0.3 (12y)
100 points!
Answer:
we can say that 0.2(18y) is equivalent to 0.3(12y).
Step-by-step explanation:
We can simplify both expressions as follows:
0.2(18y) = 3.6y
0.3(12y) = 3.6y
As we can see, both expressions simplify to the same value, 3.6y. Therefore, we can say that 0.2(18y) is equivalent to 0.3(12y)
The diagram is a plane figure with five sides. From one vertex a line was drawn to the non-consecutive vertices. What is the sum of the interior angles of the polygon?
The sum of the interior angles of the polygon is 540° by the Sum of angles property of pentagon.
We know that the diagram is a plane figure with five sides, which means it is a Pentagon (Five-sided polygon), when from one vertex a line was drawn to the non-consecutive vertices, we need to find the sum of the interior angles of the polygon:
The total of interior angles in a polygon can be found by multiplying the quantity of triangles by 180°. It is noticeable that the number of triangles is consistently two less than the number of sides.
As a result, we can conclude that the formula for determining the sum of interior angles in a convex polygon with n sides is:
S = (n - 2) × 180°
This formula provides the sum of interior angles for any polygon.
Solution: A pentagon has five sides.
Therefore, by the angle sum formula we know that:
S = (n − 2) × 180°
Here we know that n = 5 as pentagon has 5 sides,
Hence, by the Sum of angles of pentagon = (5 − 2) × 180°
S = 3 × 180°
S = 540°
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Answer:
540°
Step-by-step explanation:
1. At the time of writing this test, Ms. Shaffer was 707,472,931 seconds old. What is
this age in days? Use dimensional analysis, show all work to receive full credit.
Hint: (60 seconds = 1 minute, 60 minutes = 1 hour, 24 hours = 1 day). (6 pts)
Ms. Shaffer is approximately 8,197.42 days old.
Convert day into secondsTo convert from days to seconds, you need to multiply the number of days by the conversion factor of 86400 seconds/day. This is because there are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute, which gives us 24 x 60 x 60 = 86400 seconds in a day.
To convert seconds to days using dimensional analysis, we need to use conversion factors:
60 seconds = 1 minute
60 minutes = 1 hour
24 hours = 1 day
Hence, we can put up the calculation shown below:
707,472,931 seconds x (1 minute / 60 seconds) x (1 hour / 60 minutes) x (1 day / 24 hours)
Simplifying this expression, we get:
= 8,197.42 days
Therefore, Ms. Shaffer is approximately 8,197.42 days old.
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fix her error, please help!!!
The trigonometric ratios of sine, cosine, and tangent can be used to find the lengths of the sides of the triangle.
How do you solve the right triangles?A right triangle has one angle that measures exactly 90 degrees. Identify this angle in the triangle.
Label the sides of the triangle as the hypotenuse, adjacent, and opposite sides, relative to one of the non-right angles. The hypotenuse is always the side opposite the right angle.
We know that;
Tan 45 = 5.9/w
w = 5.9/Tan45
w = 5.9
Then;
Tan 35 = 5.9/w + x
0.7 = 5.9/5.9 + x
0.7(5.9 + x) = 5.9
4.13 + 0.7x = 5.9
x = 2.5
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What type of function is represented by the table
X= -4 -3 -2 -1
Y= 16 8 4 2
The function represented by the table is: [tex]Y = (1/2)^X[/tex].This is an example of an exponential function.
What is Algebraic function ?
An algebraic function is a function created by applying the operation of addition, subtraction, multiplication, division, and extracting the nth root
To determine the type of function represented by the table, we can look at the pattern of the values of X and Y.
As we can see from the table, as X increases by 1, Y is divided by 2. This means that the relationship between X and Y is not linear but rather exponential. Specifically, the function is of the form Y = [tex]a(1/2)^X,[/tex] where a is a constant.
To find the value of a, we can use one of the points in the table. Let's use the first point, (-4, 16):
16 = [tex]a(1/2)^(-4)[/tex]
Multiplying both sides by[tex](1/2)^(-4)[/tex]:
[tex]16 * (1/2)^4[/tex] = a
a = 16 * (1/16) = 1
Therefore, the function represented by the table is:
[tex]Y = (1/2)^X[/tex]
This is an example of an exponential function.
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Divide 14x3 − 21x2 − 7x by −7x.
i have a lot more lol
Thus, For the equation to have infinitely many solutions, the equation value is: -3x - 19 = -3x - 19.
Explain the condition for infinitely many solutions:There have been infinite solutions if the equations in the system are same.
If an equation meets certain requirements for infinite solutions, it will result in an infinite solution. The lines must coincide with a shared y-intercept in order to obtain an endless solution. To put it another way, the system would produce an endless answer if the two lines shared a single line.
The given condition is:
-3x - 19 = _x - 19
For the equation to have infinitely many solutions, both side of the equation must be equal.
Thus,
-3x - 19 = -3x - 19
such that:
-3x and -19 will get cancelled to form:
0 = 0 (which shows infinitely many solutions for the equation)
Thus, For the equation to have infinitely many solutions, the equation value is: -3x - 19 = -3x - 19.
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Graph triangle DEF with vertices D (6,2), E (6,6), and F (10,4) and its image after a dilation centered at the origin with a scale factor of 1/2. Then find the difference between the area of the image and the area of the preimage.
Answer:
The graph is below.
The area of the pre-images is 8 and the area of the image is 2. The difference is 6 [tex]units^{2}[/tex]
Step-by-step explanation:
Area of the large triangle
a = [tex]\frac{bxh}{2}[/tex] I am going to choose the vertical line to be the base because I can count the base to be 4 and the height straight up from the base to be 4.
a = [tex]\frac{4x4}{2}[/tex] = 8
I found the base for the image in the same way
a = [tex]\frac{2x2}{2}[/tex] 2
Helping in the name of Jesus.
a. Are the ratios 12 and 200
15
proportional? Explain your
reasoning.
b. Describe a situation in which thes
ratios might come up. Explain wh
it would be important to know
whether the ratios are
proportional.
The ratios of[tex]4[/tex]:[tex]5[/tex] and [tex]40[/tex]:[tex]3[/tex] are not equal, the initial ratios of [tex]12[/tex]:[tex]15[/tex] and [tex]200:15[/tex]are not proportionate.
What do you mean by ratio?When two or more numbers or values are compared, the outcome is the ratio, which is expressed as a fraction or a colon. It is used to describe how the relative magnitudes of two or more objects correlate. Ratios can be used to evaluate quantities of the same unit or of different units.
For instance, a ratio of [tex]3:1[/tex] is used to compare three units to four, while a ratio of [tex]2:1[/tex] is used to compare two units to one. Ratios can be expressed or simplified in a variety of methods.
The ratio [tex]6:9[/tex] will become[tex]2:3[/tex] if both parts are multiplied by[tex]3[/tex]. Ratios can also take the shape of decimals or percentages.
[tex]12:15 =200:15[/tex]
multiplying both the left side by[tex]3[/tex] and right side by [tex]5[/tex] we get
[tex]4:5= 40:3[/tex]
Hence it shows that they are not equal so they are not proportionate
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Mariya was asked to solve StartFraction a over negative 13 EndFraction less-than-or-equal-to negative 16 and then graph the solution. Her work is shown below.
Step
Mariya’s work
StartFraction a over negative 13 EndFraction less-than-or-equal-to negative 16
Step 1
(Negative 13) StartFraction a over negative 13 EndFraction less-than-or-equal-to negative 16 (negative 13)
Step 2
a greater-than-or-equal-to 208
Step 3
A number line going from 205 to 211. An open circle is at 208. Everything to the right of the circle is shaded.
In which step, if any, did Mariya make a mistake in her work?
An error made by Mariya resulted in incorrect graph and answer. The open circle at 208 should have been coloured to the left rather than to the right in the proper graph.
What is the inequality symbol?Mariya made a mistake in Step 1 of her work.
Step 1:
StartFraction a over negative 13 End Fraction less-than-or-equal-to negative 16
(Negative 13) StartFraction a over negative 13 End Fraction less-than-or-equal-to negative 16 (negative 13)
Mariya multiplied both sides of the inequality by -13, which would require her to flip the direction of the inequality symbol from "less than or equal to" to "greater than or equal to". However, she did not do so and continued to work with the original direction of the inequality.
The correct approach to solve the inequality would have been to multiply both sides by -13 and then flip the direction of the inequality as follows:
Step 1:
StartFraction a over negative 13 EndFraction less-than-or-equal-to negative 16
(-13) StartFraction a over negative 13 EndFraction greater-than-or-equal-to (-13) (16)
Step 2:
a greater-than-or-equal-to 208
Step 3:
A number line going from 205 to 211. An open circle is at 208. Everything to the right of the circle is shaded.
Therefore, Mariya's error in Step 1 led to an incorrect solution and an incorrect graph. The correct graph would have shaded everything to the left of the open circle at 208, instead of everything to the right.
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evaluate the integral. (remember to use absolute values where appropriate. use c for the constant of integration.) ∫ (3x^2 + 2x – 3) / (x^3 – x) dx
_____________
The integral ∫ (3x^2 + 2x – 3) / (x^3 – x) dx evaluates to:
-ln(|x|) + ln(|x - 1|) + ln(|x + 1|) + C, where C is the constant of integration.
To evaluate the integral ∫ (3x^2 + 2x – 3) / (x^3 – x) dx, follow these steps:
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