Problem 88 - Introductory and Intermediate Algebra for College Students 4th

Question

A rectangular swimming pool measures 14 feet by 30 feet. The pool is surrounded on all four sides by a path that is 3 feet wide. If the cost to resurface the path is \(\$ 2\) per square foot, what is the total cost of resurfacing the path?

Step-by-Step Solution

Answer

The total cost of resurfacing the path around the pool is $600.

Step 1: Calculate the total area of the pool with the path

To calculate the total area of the rectangle including the path, add twice the width of the path to each dimension of the pool. This is because the path surrounds the pool on both sides. So, the length of the total area is \(14 + 2(3) = 20\) feet, and the width of the total area is \(30 + 2(3) = 36\) feet. Thus, the total area (\(A_{total}\)) can be calculated by multiplying the length and width, \(A_{total} = 20 * 36 = 720\) square feet.

Step 2: Calculate the area of the pool

The area of the pool itself (\(A_{pool}\)) is the length of the pool times the width of the pool. This is given as \(A_{pool} = 14 * 30 = 420\) square feet.

Step 3: Calculate the area of the path

The area of the path (\(A_{path}\)) can be calculated by subtracting the area of the pool from the total area. This yields \(A_{path} = A_{total} - A_{pool} = 720 - 420 = 300\) square feet.

Step 4: Calculate the total cost of resurfacing

Finally, to get the total cost of resurfacing the path, multiply the cost per square foot (\(\$2\)) by the area of the path. Therefore, the total cost equals \(300 * \$2 = \$600\).