A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 15 mm and it wants to know how the area A(x) of a wafer changes when the side length x changes. Find A` (15) and explain its meaning in this situation.

Answers

Answer 1

A'(15) = 2 * 15 = 30.The meaning of A'(15) in this situation is that if the side length of the wafer changes by 1 mm, the area will change by 30 mm^2 when the side length is 15 mm.

The area A(x) of a square wafer is given by A(x) = x², where x is the side length of the wafer. The derivative of A(x) with respect to x, denoted A'(x), represents the rate of change of the area with respect to the side length. To find A'(15), we need to find the derivative of A(x) at x = 15.

A'(x) = 2x

Therefore, A'(15) = 2 * 15 = 30.

The meaning of A'(15) in this situation is that if the side length of the wafer changes by 1 mm, the area will change by 30 mm² when the side length is 15 mm. This means that for every 1 mm increase in the side length, the area of the wafer will increase by 30 mm², and for every 1 mm decrease in the side length, the area of the wafer will decrease by 30 mm².

Learn more about Calculus here: brainly.com/question/22810844

#SPJ4


Related Questions

In a right triangle, a and b are the lengths of the legs and c is the length of the hypotenuse.
If a = 6.7 feet and c = 10 feet, what is b? If necessary, round to the nearest tenth.

Answers

Answer:

7.4

Step-by-step explanation:

Use Pythagorean Theorem: [tex]a^2+b^2+c^2[/tex]

In this case we have a and c. So let's plug those into the equation:

[tex](6.7)^2+b^2=10^2[/tex]

[tex]44.9+b^2=100[/tex]

[tex]b=\sqrt{55.1}[/tex]

[tex]b=7.4[/tex]

Function A is represented by y = 4x – 3. Function B is a linear function that passes through the points shown in the table. x –1 1 3 5 y 0 4 8 12 What is the rate of change of function A? What is the rate of change of function B? Which function has a greater rate of change? Select ... Select ...

Answers

Therefore , the solution of the given problem of function comes out to be Since 4 > 2, function A has a greater rate of change than function B.

Describe Function.

The mathematics curriculum covers the study of numbers or rather their variations as well as in our environment, buildings, and both actual and imagined locations. A function presents a graph representation of the relationship between input and output quantities. A function, expression simply, is a collection of sources that, when combined, result in particular outputs by each input. There is a locale, territory, or range assigned to each job.

Here,

The rate of change of a linear function is the slope of the line.

Function A is y = 4x - 3, so the rate of change (slope) is 4.

To find the rate of change of function B, we can calculate the slope of the line passing through the points given in the table. Using the two points (1,4) and (5,12), we get:

slope = (change in y) / (change in x) = (12-4) / (5-1) = 2

So the rate of change of function B is 2.

Since 4 > 2, function A has a greater rate of change than function B.

To know more about function visit:

https://brainly.com/question/28193995

#SPJ1

What is the value of x?

Answers

Answer:

im pretty sure ia 31,61

Step-by-step explanation:

√ always together equals about 15 but as of your value of eggs that could not be possible so establishing the 6 x -5 doesn't make any difference apart from the question being so brought a part the answer would be above 30 it could be 30 to 65 anything but if you equal those balls at the bottom you get to .5 as again does 2.5 x 8 has something that equals a bunch of those 2.5 x 8 is 24.5 making 24.5 but the top one makes it al be 31.5

Answer: 10⁰

Step-by-step explanation:

We know that,

 the summation of 3 angles of a triangle is 180⁰.

now, (6x-5)⁰+(5x-5)⁰+8x⁰=180

 or, 19x-10=180⁰

 or, 19x=190⁰

 or, x= (190/19)⁰

 or, x= 10⁰

an object is thrown upward at a speed of 156 feet per second by a machine from a height of 19 feet off the ground. the height of the object after seconds can be found using the equation h= -16t^2 +156t +5. When will the height 269feet?. When will the object reach the ground?

Answers

Solving the quadratic equation, we found that the object is at a height of 269 feet when t is 1.99s and 8.55s and the object reaches the ground when t = 9.78s.

What is a quadratic equation?

Any equation in algebra that can be written in standard form:

          ax² + bx + c =0

where x stands for an unknown value, where a, b, and c stand for known values, and where a 0 is true is known as a quadratic equation.

The given equation of height h = -16t² + 156t +5

a) The time when the height is 269 feet can be found by substituting this value for h in the above equation.

            h = -16t² + 156t +5

         169 = -16t² + 156t +5

         16t² - 156t + 164 = 0

     Solving we get t = 8.55 s, 1.99s

b) The time when the object reaches the ground.

For this, we can take h = 0

   -16t² + 156t +5 = 0

  t = -0.03, 9.78

The negative value can be ignored.

Therefore solving the quadratic equation, we found that the object is at a height of 269 feet when t is 1.99s and 8.55s and the object reaches the ground when t = 9.78s.

To learn more about quadratic equations, follow the link.

https://brainly.com/question/28038123

#SPJ4

Jewel completed 12 out of 20 of her homework questions last night. What percent did she complete?​

Answers

Answer:

60%

Step-by-step explanation:

12/20 = 3/5 = 0.6 = 60%

Answer:Jewel completed 60% of her homework questions last night.Step-by-step explanation:

Hey there! To find out how much of her homework Jewel completed, all we need to do is a little math. First, let's divide the number of questions she did by the total number of questions. That's 12 divided by 20 which equals 0.6.

Now, we just need to turn that decimal into a percentage. To do that, we simply multiply 0.6 by 100, which gives us 60. And voila! Jewel completed 60% of her homework.

6.6 Midpoints and Bisectors
(6.6.4 Apply the concept of midpoint to solve real-life problems)

55. SCAVENGER HUNT Pablo is going to ask Bianca to prom by sending her on a scavenger hunt. At the end of the scavenger hunt, Pablo will be standing halfway between the gazebo and the ice cream shop in town. Where should Pablo stand?

Answers

The midpoint is given by the coordinates M ( 6 , 7.5 )

What is the midpoint of two points?

The midpoint of a line segment is a point that lies halfway between 2 points. The midpoint is the same distance from each endpoint.

Measure the distance between the two end points, and divide by 2.

Let A ( x₁ , y₁ ) be the first point

Let B ( x₂ , y₂ ) be the second point

The midpoint between A and B is M ( a , b ) where

a = ( x₁ + x₂ )/2

b = ( y₁ + y₂ ) / 2

Given data ,

Let the midpoint of gazebo and the ice cream shop in town be M ( a , b )

Now , the equation will be

The coordinates of the point Gazebo = G ( 10 , 12 )

The coordinates of the point Ice Cream = C ( 2 , 3 )

Now , midpoint between A and B is M ( a , b ) where

a = ( x₁ + x₂ )/2

b = ( y₁ + y₂ ) / 2

Substituting the values in the equation , we get

a = ( 10 + 2 ) / 2

a = 12 / 2

a = 6

b = ( 12 + 3 ) / 2

b = 15 / 2

b = 7.5

So , the coordinates of the midpoint is M ( 6 , 7.5 )

Hence , the midpoint is M ( 6 , 7.5 )

To learn more about midpoint of two points click :

https://brainly.com/question/29155265

#SPJ1

I’m confused please help

Answers

Answer:

First option: 4.9 in

Step-by-step explanation:

The area of a circle, given radius = r is computed using the formula
A = πr²

Here we are given the area and asked to find the radius r

A = 75.43 in² = πr²

So
πr² = 75.43
r² = 75.43/π
   = 24.01
   = 24 rounded to nearest integer

r = √24
  = 4.9

need help with pre cal hw

Answers

The factors of the quadratic function  x² - 2x - 4  is equal to

(x + 1 + √5)(x - 1 + √5)

What is a factor of a polynomial?

We know that if x = a is one of the roots of a given polynomial x - a = 0 is a factor of the given polynomial.

To confirm if x - a = 0 is a factor of a polynomial we replace f(x) with f(a) and if the remainder is zero then it is confirmed that x - a = 0 is a factor.

Given, The zeros of the quadratic function x² - 2x - 4  are,

(1 + √5) and (1 - √5). (As they occur in conjugate pairs).

Therefore, The factors of (x + (1 + √5))(x + (x - (1 - √5))

x² - 2x - 4 = (x + 1 + √5)(x - 1 + √5)

learn more about polynomials here :

brainly.com/question/11536910

#SPJ1

Rahela's account has an annual interest rate of 6.5% compounded semiannually. What is the annual percentage yield for Rahela's account?

Answers

Answer: The annual percentage yield for Rahela's account can be calculated by finding the effective annual interest rate, which takes into account the frequency of compounding. The effective annual interest rate is calculated as follows:

Effective annual interest rate = (1 + (6.5/2))^2 - 1 = 6.6725%

So, the annual percentage yield for Rahela's account is 6.6725%.

Step-by-step explanation:

Drag and drop the constant of proportionality into the box to match the table. If the table is not proportional, drag and drop "not proportional" into the box.

Answers

The value of proportionality is not same.So the it is not proportional.

What is Proportionality?

Any relationship that has a constant ratio is said to be proportionate. For instance, the ratio of proportionality is the average number of apples per tree, and the amount of apples in a crop is proportional to the number of trees in the orchard.

According to question:

Let

y ∝ x

y = kx , k is constant

Take x = 2, y = 3

3 = k(2)

k = 3/2 = 1.5

At x= 4, y = 7

7 = k(4)

k = 7/4 = 1.75

The value of proportionality is not same.So the it is not proportional.

To know more about Proportional visit:

brainly.com/question/28979975

#SPJ1

Simplify.
4x²+2x
x²+x-2
8x² + 4x
3x2 +10x+8

Answers

It’s the 3d question

I'm doing ixl and I don't understand what I need to do in the problem.​

Answers

Answer:

3C ÷ 21 = 63 C = 21

great question BTW

Sally saved $182 in March. Her father gave her $20 for every
$50 she saved. How much did Sally's father give her?

Answers

Answer:

$60

50×3=150

182-150=32

so Sally made 3 $50 so 20+20+20=60

determine if the statements are true or false. an even degree polynomial must be an even function. true false every polynomial of odd degree has at least one zero. true false every rational function that is not a polynomial has a vertical asymptote. a. true b. false

Answers

A rational function does not have any factors in the denominator that can be equal to 0, then it will not have a vertical asymptote.

An even degree  must be an even function is true. This is because a polynomial with an even degree has an even-numbered power, such as f(x)=x^2. This means that the function is symmetrical when its graph is reflected over the y-axis. This means that the graph will look the same if the x-values are reflected over the y-axis. For example, if f(3)=9, then f(-3)=9 as well.

Every polynomial of odd degree has at least one zero is true. This is because a polynomial with an odd degree will always have at least one x-value for which the equation is equal to 0. This is because the highest power of the equation must be odd, and the equation must cross the x-axis at least once. For example, f(x)=x^3 will have at least one x-value, such as x=0, where f(x)=0.

Every rational function that is not a polynomial has a vertical asymptote is false. A rational function that is not a polynomial may have a vertical asymptote, but it is not guaranteed. A vertical asymptote is created when the denominator of a rational equation has a factor that is equal to 0. This means that when this factor is equal to 0, the equation has an undefined value. If a rational function does not have any factors in the denominator that can be equal to 0, then it will not have a vertical asymptote.

Learn more about vertical asymptote here:

https://brainly.com/question/12191454

#SPJ4

please help me with this question thank you

Answers

The average rate of change for the given function is -72.

What Is the Average Rate of Change?

To determine the slope of a graphed function, use the average rate of change formula. Divide the change in y-values by the change in x-values to determine the average rate of change. To determine the slope of a graphed function, use the average rate of change formula. Divide the change in y-values by the change in x-values to determine the average rate of change.

The rate of the change for the interval [a,b] is given by

[tex]\frac{f(b)-f(a)}{b-a} \\[/tex]

For the given interval [3,11],

[tex]= \frac{f(11)-f(3)}{11-3} \\\\= \frac{-623+47}{8} \\\\= -72[/tex]

So, the required average rate of change is -72.

To learn more about the rate of change

https://brainly.com/question/29362528

#SPJ1

Which term does not belong?

term
coefficient
constant
slope

Answers

Answer:

not sure but slope doesn't belong

Step-by-step explanation:

A rectangular yard measuring
30
by
35ft
is bordered (and surrounded) by a fence. Inside, a walk that is
3ft
wide goes all the way along the fence. Find the area of this walk. Be sure to include the correct unit in your answer.

Answers

You can solve this by finding the area of the big rectangular yard, then subtracting the area of the small rectangular yard (inside the walkway).

You can figure out the area of the inside rectangle by subtracting the length of the walkway. (So, the original length of 35ft would become 29ft because 3ft of the walkway would be substrates from BOTH sides (35-3-3 = 29)

The answer would then be 354 ft squared (because you are finding area, and area is given in units squared)

what is the missing length of a triangle?

H= 8yd

Area= 63.6 yd

Answers

The length of the triangle is given by the equation L = 15.9 yards

What is a Triangle?

A triangle is a plane figure or polygon with three sides and three angles.

A Triangle has three vertices and the sum of the interior angles add up to 180°

Let the Triangle be ΔABC , such that

∠A + ∠B + ∠C = 180°

The area of the triangle = ( 1/2 ) x Length x Base

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

if a² + b² = c² , it is a right triangle

if a² + b² < c² , it is an obtuse triangle

if a² + b² > c² , it is an acute triangle

Given data ,

Let the length of the triangle be represented as L

Now , the equation will be

Let the base of the triangle be represented as W

Now , the value of W = 8 yards

The area of the triangle = 63.6 yards²

Now , Area of the triangle = ( 1/2 ) x Length x Base

Substituting the values in the equation , we get

Area of the triangle A = ( 1/2 ) x 8 x L

4L = 63.6

Divide by 4 on both sides of the equation , we get

L = 15.9 yards

Hence , the length of the triangle is 15.9 yards

To learn more about triangles click :

https://brainly.com/question/16739377

#SPJ1

Complete the process of solving the equation.
Fill in the missing term and select the missing description. Simplify any
fractions.
11(u-4)=-11
U-4 =-1
U=
Add 4 to both sides

Answers

I believe U would equal 3 and 4 not sure if this answers your question but I hope it did!

A ball thrown upwards hits a roof and returns back to the ground.
The upward movement is modeled by a function s=−t2+3t+4
and the downward movement is modeled by s=−2t2+t+7
, where s is the distance (in metres) from the ground and t
is the time in seconds.
Find the height of the roof from the ground.

Answers

Answer: To find the height of the roof, we need to find the point where the ball reaches the maximum height, i.e., the point where the velocity of the ball is equal to 0.

The velocity of the ball can be found by taking the derivative of the function s. The derivative of the function s = −t^2 + 3t + 4 is given by v = ds/dt = 3 − 2t, and the derivative of the function s = −2t^2 + t + 7 is given by v = ds/dt = 1 − 4t.

Setting v = 0, we get 3 − 2t = 0 and t = 3/2 seconds.

Plugging t = 3/2 seconds into the first equation, s = −t^2 + 3t + 4, we get s = −(3/2)^2 + 3(3/2) + 4 = 9/4 + 9/2 + 4 = 7 metres.

Therefore, the height of the roof from the ground is 7 metres.

Step-by-step explanation:

A scaffold has a diagonal support beam to strengthen it. When the scaffold is 15 feet high and 5 feet wide, how long must the support beam be?​

Answers

Answer:

15.81 ft

Step-by-step explanation:

Basically it's giving your 2 sides of the right angle triangle & ask you for the long side length.

15²+5²=250

√250 =15.81

The table shows the amount of money, A, in a savings account after m months.

Answers

The equations that represent the balance A after m months are given as follows:

A - 700 = 100m.A = 700 + 100m.A = 1200 + 100(m - 5).

What is a linear function?

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

In which:

The slope m represents the rate of change of the linear function.The intercept b represents the initial amount.

For this problem, we have that each month, the balance increases by 100, hence the slope m is given as follows:

m = 100.

Hence:

y = 100x + b.

When x = 5, y = 1200, hence the intercept b is given as follows:

1200 = 500 + b

b = 700.

Hence these following two equations are correct:

A - 700 = 100m.A = 700 + 100m.

Taking the fifth month as the initial month, we consider an intercept of 1200, hence the final equation is given as follows:

A = 1200 + 100(m - 5).

More can be learned about linear functions at https://brainly.com/question/24808124

#SPJ1

circular cake is 12 inches in diameter and 4 inches high. The side and top of the cake are to be covered with icing. To the nearest square inch, what is the area that needs to be iced?

Answers

The area needed to be iced is 263.76 inches².

How to find the area of the cake that will be iced?

The circular cake is 12 inches in diameter and 4 inches high. The side and top of the cake are to be covered with icing. The area of the cake that will be iced can be found as follows;

The area to to be iced is the top and the lateral area.

Therefore,

area to be iced = πr² + 2πrh

area to be iced = πr(r + 2h)

Therefore,

r = 12 / 2 = 6 inches

h = 4 inches

Hence,

area to be iced = 6π(6 + 2(4))

area to be iced = 6π(14)

area to be iced = 84π

area to be iced = 263.76 inches²

learn more on cylinder here: https://brainly.com/question/24216040

#SPJ1

Two marbles are drawn at random and without replacement from a box containing two blue marbles and three red marbles. a. List the sample points.b. Assign probabilities to the sample points.c. Determine the probability of observing each of the following events:A: { Two blue marbles are drawn. }B:{ A red and a blue marble are drawn. }C:{ Two red marbles are drawn. }

Answers

The probabilities of all the events related to drawing two marbles without replacement are discussed below.

What is meant by probability?

The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.

Given,

  Number of blue marbles B = 2

 Number of red marbles R = 3

Number of marbles drawn = 2

a) Sample points are different possible combinations.

  { R,R} , {R,B}, {B, R}, {B,B}

b) Probablitiy of  { R,R} = 3/5 * 2/4 = 6/20 = 3/10 = 0.3

    Probablitiy of {R,B} = 3/5 * 2/4 = 0.3

    Probablitiy of {B,R} = 2/5* 3/4 = 0.3

     Probablitiy of {B,B} = 2/5 * 1/4 = 0.1

c) A: { Two blue marbles are drawn. }

    Probability = 2/5 * 1/4 = 0.1

  B:{ A red and a blue marble are drawn. }

 Probability = 0.3 ( from above)

C:{ Two red marbles are drawn. }

 Probability = 0.3 (from above)

Therefore the different sample points are found and probabilities of different events are also found.

 To learn more about probability, follow the link.

https://brainly.com/question/251701

#SPJ4

Suppose a function f(x) is defined on the domain [-8,4]. If we define a new function g(x) by g(x) = f(-2x), then what is the domain of g(x)? Express your answer in interval notation

Answers

The required domain of the function If g(x) = f(-2x) is [-16, 8] in interval notation.

How to find domain of function?

The domain of g(x) is the set of all x for which the function g(x) is defined. If g(x) = f(-2x), then the domain of g(x) is the same as the domain of -2x, subject to the restriction that f(x) is defined for the corresponding values of x.

According to question:

In interval notation, the domain of -2x is (-∞, ∞).

However, since f(x) is defined only on the interval [-8, 4], the domain of g(x) must be restricted to values of x that correspond to values of -2x within the interval [-8, 4].

To find the corresponding values of x, we need to solve the equation -2x = t for x, where t is a number in the interval [-8, 4].

Solving for x, we get x = -t/2. So, the domain of g(x) is the set of all x such that -t/2 is in the interval [-8, 4], or equivalently, t is in the interval [-16, 8]. In interval notation, this is [-16, 8].

So, the domain of g(x) is [-16, 8] in interval notation.

To know more about Domain visit:

brainly.com/question/28135761

#SPJ4

Compare A and B in three​ ways, where A= 51,102 is the number of deaths due to a deadly disease in the United States in 2005 and B= 17,056 is the number of deaths due to the same disease in the United States in 2009.

Answers

Answer:

Step-by-step explanation:

Here are three ways to compare A and B, where A= 51,102 is the number of deaths due to a deadly disease in the United States in 2005 and B= 17,056 is the number of deaths due to the same disease in the United States in 2009:

Magnitude: A is approximately three times larger than B. This means that the number of deaths due to the disease was significantly higher in 2005 than in 2009.

Trend: There was a significant decrease in the number of deaths due to the disease between 2005 and 2009. The number of deaths decreased by approximately 67%, from 51,102 in 2005 to 17,056 in 2009.

Impact: Despite the significant decrease in the number of deaths, the disease still caused a significant number of deaths in both 2005 and 2009. This highlights the ongoing importance of disease prevention and treatment efforts to reduce the impact of the disease on public health.

The population in a certain town is increasing linearly each year. The population at time t=4 is 1580 and at time t=7 is 2000, where t is the number of years after 1990.

If P(t) is the population at time t, which of these equations correctly represents this situation?

Answers

We can use the two points given to find the slope of the line (which represents the population growth) and then use the point-slope form of the equation of a line to write an equation.

The population at time t=4 is 1580, which means P(4) = 1580. The population at time t=7 is 2000, which means P(7) = 2000. We can use these two points to find the slope:

slope = (P(7) - P(4)) / (7 - 4) = (2000 - 1580) / 3 = 140

So the slope of the line representing population growth is 140. Now we can use the point-slope form of the equation of a line to write an equation:

P - 1580 = 140(t - 4)

Simplifying and rearranging, we get:

P = 140t - 380

So the correct equation that represents the situation is:

P(t) = 140t - 380.

How many meters are in 956 centimeters

Answers

Answer:

956 centimeters is equal to 9.56 meters.

How many meters are in 956 centimeters?

Well, first, we must understand the conversion from meters to centimeters and centimeters to meters.

1m=100cm and 1cm=0.01m. So multiply by 100 if converting from m to cm and divide by 100 if converting from cm to m.

In this problem, we're looking at cm to m.

Take 956cm and divide that by 100. You'll get 9.56m. That's your answer.

Hope this helped!

Please Help I need this ASAP will award Brainliest if correct

Answers

The center of dilation is (1, -4).

What is a dilation?

Dilation is the process of resizing or transforming an object. It is a transformation that makes the objects smaller or larger with the help of the given scale factor. The new figure obtained after dilation is called the image and the original image is called the pre-image.

From the given graph,

Vertices of RSTV are R(1, -4), S(2, 0), T(1, 2) and V(-2, 1).

Vertices of R'S'T'V' are R'(1, -4), S'(4.5, 2), T'(2, 5) and V'(-2.5, 3.5).

Here, the center of dilation is (1, -4).

The transformation is enlargement by scale factor 1.5

Therefore, the center of dilation is (1, -4).

To learn more about the dilation visit:

https://brainly.com/question/13176891.

#SPJ1

6.9.4 Journal: Similar circles6.9.4 Journal: Similar Circles

Journal

Geometry Sem 1

Points Possible:20

Name:


Date:



Scenario: Prove That All Circles Are Similar
Instructions
View the video found on page 1 of this Journal activity.
Using the information provided in the video, answer the questions below.
Show your work for all calculations.
The Students' Conjectures:The two students have different methods for proving that all circles are similar.
1. Complete the table to summarize each student's conjecture about how to solve the problem. (2 points: 1 point for each row of the chart)
Classmate Conjecture
John



Teresa





Evaluate the Conjectures:
2. Intuitively, does it make sense that all circles are similar? Why or why not? (1 point)








Construct the Circles:
3. Draw two circles with the same center. Label the radius of the smaller circle r1 and the radius of the larger circle r2. Use the diagram you have drawn for questions 3 – 10. (2 points)
















4. In your diagram in question 3, draw an isosceles right triangle inscribed inside the smaller circle. Label this triangle ABC. (1 point)
5. What do you know about the hypotenuse of △ABC? (2 points)




6. In your diagram in question 3, extend the hypotenuse of △ABC so that it creates the hypotenuse of a right triangle inscribed in the larger circle. Add point Y to the larger circle so it is equidistant from X and Z. Then complete isosceles triangle XYZ. (1 point)
7. What do you know about the hypotenuse of △XYZ? (2 points)




8. How does △ABC compare with △XYZ? Explain your reasoning. (2 points)








9. Use the fact that △ABC ≈ △XYZ to show that the ratio of the radii is a constant. (2 points)








Making a Decision
10. Who was right, Teresa or John? (1 point)




Further Exploration:
11. What is the circumference of the circle that circumscribes a triangle with side lengths 3, 4, and 5? (4 points)











Transcript: Similar Circles
The video begins with a young woman talking in front of a blank screen.

Audio:

I'm Teresa. My friend John and I need to prove that all circles are similar. It seems obvious, right?! Of course they're similar, they’re all circles!

[Many circles of different sizes and colors pop up onto the screen.]

Um, this is making me a little dizzy.

[The circles disappear.]

But we do need to prove that all circles are mathematically similar.

Here’s the way John looks at it: Remember what we learned about similar triangles?

[Two triangles appear on the screen. One is small and the other is large.]

We can take one triangle, and move it on top of another triangle.

[The small triangle is placed on top of the large triangle.]

Then, we dilate it to show that they are similar. Like that.

[The small triangle is dilated to the size of the large triangle.]

John says we can do the same thing with circles.

[Two circles appear on the screen. One is small and the other is large.]

Take any two circles, and move them so that they have the same center.

[The small circle is moved on top of the large circle.]

Then, you can dilate or contract the circles until they are the same size.

[The small circle is dilated to the size of the large circle.]

Taa-daa! The circles are similar. I have another way to prove it.

[Two triangles appear on the screen. The small triangle has sides of length 2, 2, and 3 and the large triangle has sides of length 6, 6, and 9.]

We also know that triangles are similar if all of their corresponding sides have the same ratio.

[The corresponding sides of the triangle are highlighted. On-screen text: 2 over 6 equals 2 over 6 which equals 3 over 9 which equals 1 over 3 Similar!]

Well, the same idea should also work with circles.

[Two circles appear on the screen. One circle is small and the other circle is large.]

If the corresponding parts of two circles have the same ratio, then the circles must be similar. And lucky for us, everything about a circle can be described with its radius!

[Beneath the small circle is written "equals r sub 1." Inside the large circle is written "equals r sub 2." On-screen text: Diameter equals 2r, Circumference equals 2pi r, and Area equals pi r squared.]

So, if the radii of these circles have a constant ratio, then the circles are similar.

[On-screen text: If r sub 1 over r sub 2 equals k, a constant, then the circles are similar.]

What's more, I think I can prove all this by using inscribed triangles. But I need your help.

[A triangle is inscribed in each of the circles using the diameter of the circles as their bases.]

Can we actually use inscribed right triangles to show that all circles are similar?

Answers

Step-by-step explanation:

Classmate Conjecture

John | All circles are similar if they have the same center and can be dilated or contracted to the same size.

Teresa | All circles are similar if their corresponding radii have a constant ratio.

Evaluate the Conjectures:

2. Yes, it makes intuitive sense that all circles are similar because they all have the same shape and form.

Construct the Circles:

3. [Diagram not provided]

[Diagram not provided]

The hypotenuse of △ABC is the diameter of the smaller circle.

[Diagram not provided]

The hypotenuse of △XYZ is the diameter of the larger circle.

The two triangles, △ABC and △XYZ, are similar because they are both isosceles right triangles with the same angle measures.

The ratio of the lengths of their sides are equal, therefore the ratio of their radii is a constant.

Making a Decision:

10. Both John and Teresa were right as all circles are similar if they have the same center and can be dilated or contracted to the same size (John's method) and also if their corresponding radii have a constant ratio (Teresa's method).

Further Exploration:

11. The circumference of the circle that circumscribes a triangle with side lengths 3, 4, and 5 can be found using the Pythagorean theorem to find the diameter of the circle, which is equal to the sum of the lengths of the three sides. The diameter is equal to 5, so the circumference is equal to 2 * pi * (5 / 2) = 5 * pi

Please provide all information when asking a question

ANSWER -

Classmate Conjecture
John
Takes two circles, moves them so they have the same center, then dilates or contracts them until they are the same size.

Teresa
Uses the idea that triangles are similar if all of their corresponding sides have the same ratio. Proves that all circles are similar by showing that if the radii of the two circles have a constant ratio, then the circles are similar.

Evaluate the Conjectures:
2. Intuitively, does it make sense that all circles are similar? Why or why not? (1 point)
Yes, it makes sense that all circles are similar because they have the same basic shape and structure, regardless of their size.

Construct the Circles:
3. Draw two circles with the same center. Label the radius of the smaller circle r1 and the radius of the larger circle r2. Use the diagram you have drawn for questions 3 – 10. (2 points)
[Diagram not provided]

In your diagram in question 3, draw an isosceles right triangle inscribed inside the smaller circle. Label this triangle ABC. (1 point)
[Diagram not provided]
What do you know about the hypotenuse of △ABC? (2 points)

The hypotenuse of △ABC is equal to the diameter of the smaller circle.
In your diagram in question 3, extend the hypotenuse of △ABC so that it creates the hypotenuse of a right triangle inscribed in the larger circle. Add point Y to the larger circle so it is equidistant from X and Z. Then complete isosceles triangle XYZ. (1 point)
[Diagram not provided]

What do you know about the hypotenuse of △XYZ? (2 points)
The hypotenuse of △XYZ is equal to the diameter of the larger circle.
How does △ABC compare with △XYZ? Explain your reasoning. (2 points)

△ABC and △XYZ are similar triangles because they have the same shape, but different sizes. This is because the hypotenuse of △ABC is equal to the diameter of the smaller circle, and the hypotenuse of △XYZ is equal to the diameter of the larger circle.

Use the fact that △ABC ≈ △XYZ to show that the ratio of the radii is a constant. (2 points)
Since △ABC and △XYZ are similar, we can write the following proportion:
r1 / r2 = AB / XY = BC / YZ

So, the ratio of the radii, r1 / r2, is a constant, which means that all circles with the same center are similar.

Making a Decision
10. Who was right, Teresa or John? (1 point)
Both Teresa and John were right.

Further Exploration:
11. What is the circumference of the circle that circumscribes a triangle with side lengths 3, 4, and 5? (4 points)
[Solution not provided]
Other Questions
the existence of a miss black deaf america pageant is just one example of how far the black deaf community has come in providing visibility and role models, uniting and advocating for a common cause, and showcasing leadership.a. trueb. false A small country emits 92,000 kilotons of carbon dioxide per year. In a recent global agreement, the country agreed to cut its carbon emissions by 5% per year for the next 15 years. In the first year of the agreement, the country will keep its emissions at 92,000 kilotons and the emissions will decrease 5% in each successive year. How many total kilotons of carbon dioxide would the country emit over the course of the 15 year period, to the nearest whole number? (01.04 LC)A demand curve that shifts to the left indicates a(n) (5 points)O increase in demandO decrease in demandO increase in supplyO decrease in supply there is a deep trench in the sea floor off the marianas volcanic arc of explosive, andesitic, ring of fire volcanoes in the south pacific, but the water is not deep off the coast of oregon and washington near mt. st. helens and the olympic, because: A company manufactures computer memory chips as circular silicon wafers with a diameter of 10 inches. The wafers are cut into different sized sectors. Match each wafer's central angle to the area of that sector. (PLEASE HEEEEEEEEELP!!!!!) suppose that you have the following declaration. enum cars {ford, gm, toyota, honda}; cars domesticcars Which eoc configuration aligns with the on-scene incident? Which of the following is the square of a binomial Write 5 verbs that end in AR, 5 that end inER and five that end in IRConjugate 1 verbs in the past tensethat ends in AR, one in ER and one in IRWrite 1 sentence with each conjugated verb the distinctive architecture of the internet is wholly consistent with an expansive and robust conception of free speech rights. (true or false) How and when were labor unions created? Which interval notation represents the set of all number greater than or equal to 1and less than 6? What happens after a hormone has exerted its effects? If a regular octahedron has a surface area of 32 square inches what is the surface area of each face? despite americans' ability to vote for many top officials, some critics assert that monied interests in campaign finance have come to outweigh the voice of the people in the country's governance and policymaking. in so doing, they suggest that the united states is orWatch a videoFind the cosine of U.284553UVTSimplify your answer and write it as a proper fraction, improper fraction, or whole number. temporalis muscles and their attachments describe the origin and the insertion of this muscle. how is this structure linked to the function? Find the value of each variable on the triangle One side x one side y one side 13 and the measurement is 45 degrees what is the spacing between the (211) planes of lead? would you expect to see this plane diffract in an xrd experiment? why or why not? lead has a fcc crystal structure with an atomic radius of 0.175 nm.