Suppose you take a straight piece of wire measuring 20 cm and cut it to make two pieces. You then take one piece and make it into a square, and the other piece and make it into a circle, like in the following picture: 20 cm 20-x а. Write a function, 4), that gives the total area of the circle and the square you make in this scenario. b. What is the domain of A(.x)?
Suppose you take a straight piece of wire measuring 20 cm and cut it to make two pieces. You then take one piece and make it into a square, and the other piece and make it into a circle, like in the following picture: 20 cm 20-x а. Write a function, 4), that gives the total area of the circle and the square you make in this scenario. b. What is the domain of A(.x)?
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
### Questions
a. Write a function, \( A(x) \), that gives the total area of the circle and the square you make in this scenario.
b. What is the domain of \( A(x) \)?
### Detailed Explanation
To solve this problem, we first need to use the lengths of the wire pieces for both the circle and the square.
1. **Dividing the Wire**:
- Let \( x \) cm be the length of the wire used to form the circle.
- Then, \( 20 - x \) cm will be the length of the wire used to form the square.
2. **Forming the Circle**:
- The circumference of the circle is \( x \) cm.
- Using the formula for the circumference of a circle, we have:
\[
2\pi r = x
\]
- Solving for the radius \( r \), we get:
\[
r = \frac{x}{2\pi}
\]
- The area of the circle, \( A_{\text{circle}} \), is given by:
\[
A_{\text{circle}} = \pi r^2 = \pi \left( \frac{x}{2\pi} \right)^2 = \frac{x^2}{4\pi}
\]
3. **Forming the Square**:
- The perimeter of the square is \( 20 - x \) cm.
- Each side of the square, \( s \), is:
\[
s = \frac{20 - x}{4}
\]
- The area of the square, \( A_{\text{square}} \), is given by:
\[
A_{\text{square}} = s^2 = \left( \frac{20 - x}{4} \right)^2 = \frac{(20 - x)^2}{16}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe91e659-0274-4af0-b893-ea272a52ff02%2Fc46b3bd7-5d70-4695-b9ba-8acb3a92ee87%2Fefiw2mj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Description
Suppose you take a straight piece of wire measuring 20 cm and cut it to make two pieces. You then take one piece and make it into a square, and the other piece and make it into a circle, like in the following picture:
### Questions
a. Write a function, \( A(x) \), that gives the total area of the circle and the square you make in this scenario.
b. What is the domain of \( A(x) \)?
### Detailed Explanation
To solve this problem, we first need to use the lengths of the wire pieces for both the circle and the square.
1. **Dividing the Wire**:
- Let \( x \) cm be the length of the wire used to form the circle.
- Then, \( 20 - x \) cm will be the length of the wire used to form the square.
2. **Forming the Circle**:
- The circumference of the circle is \( x \) cm.
- Using the formula for the circumference of a circle, we have:
\[
2\pi r = x
\]
- Solving for the radius \( r \), we get:
\[
r = \frac{x}{2\pi}
\]
- The area of the circle, \( A_{\text{circle}} \), is given by:
\[
A_{\text{circle}} = \pi r^2 = \pi \left( \frac{x}{2\pi} \right)^2 = \frac{x^2}{4\pi}
\]
3. **Forming the Square**:
- The perimeter of the square is \( 20 - x \) cm.
- Each side of the square, \( s \), is:
\[
s = \frac{20 - x}{4}
\]
- The area of the square, \( A_{\text{square}} \), is given by:
\[
A_{\text{square}} = s^2 = \left( \frac{20 - x}{4} \right)^2 = \frac{(20 - x)^2}{16}
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