e owners of the rectangular swimming pool in the illustration want to surround the pool with a crushed-stone border of uniform width. They have enough stone to cover 90 square meters. How wide ould they make the border? (Hint: The area of the larger rectangle minus the area of the smaller is the area of the border. Assume a = 29 m and b = 14 m.) 3D b+ 2w a + 2w b meters a meters
e owners of the rectangular swimming pool in the illustration want to surround the pool with a crushed-stone border of uniform width. They have enough stone to cover 90 square meters. How wide ould they make the border? (Hint: The area of the larger rectangle minus the area of the smaller is the area of the border. Assume a = 29 m and b = 14 m.) 3D b+ 2w a + 2w b meters a meters
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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![### How Wide Should the Border Be?
The owners of the rectangular swimming pool in the illustration want to surround the pool with a crushed-stone border of uniform width. They have enough stone to cover 90 square meters. How wide should they make the border?
**Hint**: The area of the larger rectangle minus the area of the smaller is the area of the border. Assume a = 29 meters and b = 14 meters.
**Equation to Solve:**
\[ w = \_\_\_\_\_\_ \text{ meters} \]
### Diagram Explanation:
The diagram shows a top view of a rectangular swimming pool with dimensions:
- Length (a) = 29 meters
- Width (b) = 14 meters
There is a border surrounding the pool of uniform width, denoted as \( w \). The overall dimensions of the larger rectangle (pool plus border) are:
- Length = \( a + 2w \)
- Width = \( b + 2w \)
The area of the smaller (pool) rectangle is \( \text{a} \times \text{b} \).
The area of the larger rectangle (pool plus border) is \( (a + 2w) \times (b + 2w) \).
Let’s calculate the area of the borders:
\[ (\text{area of larger rectangle}) - (\text{area of smaller rectangle}) = 90 \, \text{square meters}\]
Parameters:
- \(a = 29 \, \text{m}\)
- \(b = 14 \, \text{m}\)
### Calculation Steps:
1. Write the formula for the larger rectangle area:
\[ (29 + 2w)(14 + 2w) \]
2. Subtract the area of the pool:
\[ (29 + 2w)(14 + 2w) - 29 \times 14 = 90 \]
3. Solve for \( w \):
First, expand the left-hand side:
\[ (29 + 2w)(14 + 2w) - 406 = 90 \]
Then, we have:
\[ 406 + 58w + 4w^2 - 406 = 90 \]
\[ 4w^2 + 58w = 90 \]
Finally, simplify and solve the quadratic equation:
\[ 4w^2 + 58w - 90](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7d8b63c-f6b2-4537-9353-75fc012a2847%2F08078e94-9bd8-4f32-a8ca-cb171a11f03b%2Fg0g1ic6_processed.png&w=3840&q=75)
Transcribed Image Text:### How Wide Should the Border Be?
The owners of the rectangular swimming pool in the illustration want to surround the pool with a crushed-stone border of uniform width. They have enough stone to cover 90 square meters. How wide should they make the border?
**Hint**: The area of the larger rectangle minus the area of the smaller is the area of the border. Assume a = 29 meters and b = 14 meters.
**Equation to Solve:**
\[ w = \_\_\_\_\_\_ \text{ meters} \]
### Diagram Explanation:
The diagram shows a top view of a rectangular swimming pool with dimensions:
- Length (a) = 29 meters
- Width (b) = 14 meters
There is a border surrounding the pool of uniform width, denoted as \( w \). The overall dimensions of the larger rectangle (pool plus border) are:
- Length = \( a + 2w \)
- Width = \( b + 2w \)
The area of the smaller (pool) rectangle is \( \text{a} \times \text{b} \).
The area of the larger rectangle (pool plus border) is \( (a + 2w) \times (b + 2w) \).
Let’s calculate the area of the borders:
\[ (\text{area of larger rectangle}) - (\text{area of smaller rectangle}) = 90 \, \text{square meters}\]
Parameters:
- \(a = 29 \, \text{m}\)
- \(b = 14 \, \text{m}\)
### Calculation Steps:
1. Write the formula for the larger rectangle area:
\[ (29 + 2w)(14 + 2w) \]
2. Subtract the area of the pool:
\[ (29 + 2w)(14 + 2w) - 29 \times 14 = 90 \]
3. Solve for \( w \):
First, expand the left-hand side:
\[ (29 + 2w)(14 + 2w) - 406 = 90 \]
Then, we have:
\[ 406 + 58w + 4w^2 - 406 = 90 \]
\[ 4w^2 + 58w = 90 \]
Finally, simplify and solve the quadratic equation:
\[ 4w^2 + 58w - 90
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