A pop-up sprinkler sprays water in a semicircular area, as modeled in the figure below. What is the approximate area of lawn watered by the sprinkler?
A pop-up sprinkler sprays water in a semicircular area, as modeled in the figure below. What is the approximate area of lawn watered by the sprinkler?
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem Statement:**
A pop-up sprinkler sprays water in a semicircular area, as modeled in the figure below. What is the approximate area of lawn watered by the sprinkler?
**Figure Explanation:**
The diagram shows a semicircle with a radius of 4 yards, centered on the sprinkler. The area covered by the sprinkler is represented by dashed lines emanating from the sprinkler, indicating the spread of water within the semicircular region.
**Multiple Choice Options:**
- 50.3 square yards
- 25.1 square yards
- 6.3 square yards
- 12.6 square yards
**Solution Explanation:**
To find the area of the semicircular region watered by the sprinkler, use the formula for the area of a circle and then divide by 2, since it's a semicircle:
\[ \text{Area of circle} = \pi r^2 \]
Here, \( r = 4 \) yards.
\[ \text{Area of circle} = \pi \times 4^2 = 16\pi \]
\[ \text{Area of semicircle} = \frac{16\pi}{2} = 8\pi \]
Approximating \( \pi \) as 3.14:
\[ \text{Area of semicircle} \approx 8 \times 3.14 = 25.12 \text{ square yards} \]
The approximate area of the lawn watered by the sprinkler is **25.1 square yards**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40ac1924-da82-497b-85f4-398551157699%2Ff7ddc31c-5407-4350-9772-fd60ff84e7ff%2Fjyecwqp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A pop-up sprinkler sprays water in a semicircular area, as modeled in the figure below. What is the approximate area of lawn watered by the sprinkler?
**Figure Explanation:**
The diagram shows a semicircle with a radius of 4 yards, centered on the sprinkler. The area covered by the sprinkler is represented by dashed lines emanating from the sprinkler, indicating the spread of water within the semicircular region.
**Multiple Choice Options:**
- 50.3 square yards
- 25.1 square yards
- 6.3 square yards
- 12.6 square yards
**Solution Explanation:**
To find the area of the semicircular region watered by the sprinkler, use the formula for the area of a circle and then divide by 2, since it's a semicircle:
\[ \text{Area of circle} = \pi r^2 \]
Here, \( r = 4 \) yards.
\[ \text{Area of circle} = \pi \times 4^2 = 16\pi \]
\[ \text{Area of semicircle} = \frac{16\pi}{2} = 8\pi \]
Approximating \( \pi \) as 3.14:
\[ \text{Area of semicircle} \approx 8 \times 3.14 = 25.12 \text{ square yards} \]
The approximate area of the lawn watered by the sprinkler is **25.1 square yards**.
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