Problem 47 - Thinking Mathematically

Question

A school playground is in the shape of a rectangle 400 feet long and 200 feet wide. If fencing costs \(\$ 14\) per yard, what will it cost to place fencing around the playground?

Step-by-Step Solution

Answer

The cost to place fencing around the playground is \$5600

Step 1: Conversion of units

As 1 yard is equivalent to 3 feet, divide the length and the width of the playground by 3. Hence the new dimensions become \( \frac{400}{3} = 133.33 \) yards (length) and \( \frac{200}{3} = 66.67 \) yards (width).

Step 2: Compute the perimeter

A rectangle has its opposite sides equal. So, the total fencing required (or the perimeter of the rectangle) is the sum of all the sides. Thus, the perimeter \( P \) is given as \( P = 2 * (length + width) = 2 * (133.33 + 66.67) = 400 \) yards

Step 3: Calculate the fencing cost

Multiply the total length of the fence (perimeter) by the cost per yard to get the total cost. Thus, the cost \( C \) is \( C = cost\_per\_yard * P = 14 * 400 = \$5600 \)

Key Concepts

Unit Conversion

When dealing with measurements, understanding unit conversion is crucial. In this problem, we're converting the dimensions of a playground from feet to yards.
For every 1 yard, there are 3 feet. This conversion is necessary because the cost of the fencing is given in yards.

• To convert from feet to yards, divide the number of feet by 3.
• In this case, the length of 400 feet becomes \( \frac{400}{3} = 133.33 \) yards.


• The width of 200 feet becomes \( \frac{200}{3} = 66.67 \) yards.

This simple conversion shows the need to match units when performing any calculation, ensuring accuracy and preventing any misunderstanding in measurements.

Perimeter Calculation

The perimeter of any polygon is the total length around it. For a rectangle, the formula to calculate the perimeter \( P \) is:

• \( P = 2 \times ( \text{length} + \text{width} ) \)
• In this exercise, plug in the converted dimensions: \( P = 2 \times (133.33 + 66.67) \)
• That results in a perimeter of 400 yards.

Calculating the perimeter is essential for determining the amount of fencing required, as it represents the total boundary that needs to be enclosed. Remember, perimeter calculations for rectangles are one of the most common real-world applications in geometry.

Cost Estimation

Once the perimeter is known, cost estimation becomes straightforward. This involves multiplying the total perimeter by the cost per unit to find the total expense.

• Given that the perimeter is 400 yards and the cost per yard of fencing is \\(14, the total cost calculation is simply:
• \( C = \text{cost_peryard} \times P = 14 \times 400 \)
• This results in a total fencing cost of \\)5600.

Cost estimation is an essential skill in geometry, helping determine budget requirements for projects and ensuring cost-effective decisions are made.

Rectangular Dimensions

Recognizing the properties of a rectangle is fundamental in geometry. A rectangle is characterized by:

• Opposite sides being equal in length.
• Right angles at each corner.
• The formula for perimeter as \( 2 \times ( \text{length} + \text{width} ) \).

In real-life applications, understanding dimensions allows one to make needed transformations, like converting between units or adjusting calculations based on different materials or measurements. This basic yet essential shape allows for straightforward calculations in many practical tasks, including constructing spaces and enclosing areas with fencing, as shown in this exercise.