### 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 metersThere 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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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)

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ChapterP: Prerequisites: Fundamental Concepts Of Algebra

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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

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