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
![**Problem 13:**
**Instruction:** Write an expression for the shaded area in terms of \( k \), assuming that the shaded portion on the right-hand side is a semi-circle.
**Diagram Explanation:**
The image shows a rectangle attached to a semi-circle on its right side. The dimensions of the rectangle are given as 20 (length) and 10 (width). The shaded area includes a semi-circular portion as well as strips along the top and bottom of the rectangle with uniform width \( k \).
- The horizontal strips have a width \( k \).
- The semi-circle has a radius equal to half the width of the rectangle plus the width of the strip \( k \), which is \( 5 + k \).
**Finding the Shaded Area:**
1. **Calculate the area of the semi-circle:**
The semi-circle's radius is \( r = \frac{10}{2} + k = 5 + k \).
The area of the full circle would be \( \pi (5 + k)^2 \). Therefore, the area of the semi-circle is:
\[
\text{Area of semi-circle} = \frac{1}{2} \pi (5 + k)^2
\]
2. **Calculate the area of the horizontal strips:**
Each strip has dimensions 20 (length) by \( k \) (width).
\[
\text{Area of both strips} = 2 \times (20 \times k) = 40k
\]
3. **Total Shaded Area:**
Add the area of the semi-circle to the area of the strips to get the total shaded area:
\[
\text{Total shaded area} = \frac{1}{2} \pi (5 + k)^2 + 40k
\]
Thus, the expression for the shaded area in terms of \( k \) is:
\[
\frac{1}{2} \pi (5 + k)^2 + 40k
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92ab3c20-a3cd-4c1e-8775-52ac5ed4d4cf%2F7767ee3a-efca-4b96-a840-740d0c2bdf27%2F7kuuqrn_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 13:**
**Instruction:** Write an expression for the shaded area in terms of \( k \), assuming that the shaded portion on the right-hand side is a semi-circle.
**Diagram Explanation:**
The image shows a rectangle attached to a semi-circle on its right side. The dimensions of the rectangle are given as 20 (length) and 10 (width). The shaded area includes a semi-circular portion as well as strips along the top and bottom of the rectangle with uniform width \( k \).
- The horizontal strips have a width \( k \).
- The semi-circle has a radius equal to half the width of the rectangle plus the width of the strip \( k \), which is \( 5 + k \).
**Finding the Shaded Area:**
1. **Calculate the area of the semi-circle:**
The semi-circle's radius is \( r = \frac{10}{2} + k = 5 + k \).
The area of the full circle would be \( \pi (5 + k)^2 \). Therefore, the area of the semi-circle is:
\[
\text{Area of semi-circle} = \frac{1}{2} \pi (5 + k)^2
\]
2. **Calculate the area of the horizontal strips:**
Each strip has dimensions 20 (length) by \( k \) (width).
\[
\text{Area of both strips} = 2 \times (20 \times k) = 40k
\]
3. **Total Shaded Area:**
Add the area of the semi-circle to the area of the strips to get the total shaded area:
\[
\text{Total shaded area} = \frac{1}{2} \pi (5 + k)^2 + 40k
\]
Thus, the expression for the shaded area in terms of \( k \) is:
\[
\frac{1}{2} \pi (5 + k)^2 + 40k
\]
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