Problem 66 - Thinking Mathematically

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 is \$600.

Step 1: Calculate the total area

First, calculate the total area, including the pool and the path that surrounds it. The path is 3 feet wide on all sides, so add this to all sides of the pool. This makes the large rectangle dimensions 14 feet + 3 feet + 3 feet = 20 feet (width) and 30 feet + 3 feet + 3 feet = 36 feet (length). The area for a rectangle is calculated by multiplying the width by the length. So, the total area is \(20 \, feet \times 36 \, feet = 720 \, square \, feet\).

Step 2: Calculate the area of the pool

The area of the rectangular pool is calculated by multiplying its width by its length, i.e., \(14 \, feet \times 30 \, feet = 420 \, square \, feet\).

Step 3: Calculate the area of the path

Now, subtract the area of the pool from the total area calculated in step 1. This will give us the area of the path alone. So, the area of the path is \( 720 \, square \, feet - 420 \, square \, feet = 300 \, square \, feet\) .

Step 4: Calculate the total cost to resurface the path

Finally, to find the total cost to resurface the path, multiply the area of the path by the cost per square foot. \(300 \, square \, feet \times \$2 = \$600\)

Key Concepts

Understanding Area Calculation

When tackling geometry problems involving area calculation, it's important to first grasp the core components. In this exercise, we are essentially calculating the areas of two rectangles: the larger rectangle inclusive of the pool and path, and the smaller rectangle which is just the pool itself.
To calculate the area of a rectangle, you use the formula: - \(\text{Area} = \text{width} \times \text{length}\)- For the pool itself, the dimensions are 14 feet by 30 feet, which means its area is \(14 \, \text{feet} \times 30 \, \text{feet} = 420 \, \text{square feet}\).
For the larger rectangle (the pool plus the path), the width and length both increase by twice the path's width (since the path is on all sides). Hence, the area calculation involves: \(20 \, \text{feet} \times 36 \, \text{feet} = 720 \, \text{square feet}\). Understanding how to effectively use area formulas is key to finding the solution to problems like these.

Calculating Cost Estimation

Cost estimation in geometry problems involves calculating the expenses based on the physical dimensions of a given area. Here, we are determining how much it will cost to resurface the path around the pool by knowing the area and the cost per square foot.
Once you've calculated the area of the path, you multiply it by the cost per square foot for the resurfacing. Using the formula for cost, you have:

• \(\text{Cost} = \text{Area} \times \text{Cost per square unit}\)

Given that the path's area equals 300 square feet and the cost of resurfacing is \( \\(2 \, \text{per square foot} \), the total cost is \(300 \, \text{square feet} \times \\)2 = \$600\). This offers a practical and direct way to calculate how budget impacts geometric shapes in real-world scenarios.

Exploring Rectangular Pool Dimensions

Rectangular pool dimensions are a critical aspect of numerous geometric calculations. The exercise provided requires understanding how changes in dimensions affect total area and related calculations.
A rectangular shape is defined by its width and length, in this case, the pool measures 14 feet by 30 feet. Knowing these dimensions allows us to calculate the area, which informs subsequent calculations such as for surrounding paths. Converting dimensional changes into calculations involves simply adding values to the original dimensions for extensions like a path.
Here, the wider path effectively extends dimensions from 14 to 20 feet and 30 to 36 feet for width and length respectively. This forms the basis for another essential area calculation of a larger encompassing rectangle. Recognizing how these changes affect other components is vital for clear, precise problem-solving in geometry.