Problem 48 - Thinking Mathematically

Question

A rectangular field is 70 feet long and 30 feet wide. If fencing costs \(\$ 8\) per yard, how much will it cost to enclose the field?

Step-by-Step Solution

Answer

The cost to enclose the field is \(\$ 533.28\).

Step 1: Converting Dimensions to Yards

The length and width of the field are given in feet. Convert these dimensions to yards by dividing each of them by the number of feet in a yard which is 3. \[Length = 70 \, ft = \frac{70}{3} \, yd \approx 23.33 \, yd\] \[Width = 30 \, ft = \frac{30}{3} \, yd = 10 \, yd\]

Step 2: Calculate the Perimeter

Perimeter of a rectangle is calculated as twice the sum of its length and width. \[Perimeter = 2*(Length + Width) = 2*(23.33 \, yd + 10 \, yd) = 2*33.33 \, yd = 66.66 \, yd\]

Step 3: Calculate the Total Cost

The cost of fencing is calculated by multiplying the total length (the perimeter in this case) by the cost per yard. \[Total Cost = Perimeter * Cost \, per \, yard = 66.66 \, yd * \$ 8 \,/ \, yd = \$ 533.28\]

Key Concepts

Rectangular Field

When dealing with a rectangular field, it is important to understand its basic geometry. A rectangle has two pairs of parallel sides: a length and a width. In our example, the rectangular field measures 70 feet in length and 30 feet in width. The field is straightforwardly a two-dimensional shape, which is key to calculating its perimeter and eventually the cost to enclose it. To determine where to place a fence, consider the entire boundary surrounding the field, which is described as the perimeter. Knowing the dimensions of the field is crucial for further calculations because they directly correlate with the total length of fencing required.

Unit Conversion

In real-world applications, measurements are not always in the units we need them to be. Converting between units is essential for accurate calculations. Here, the field's dimensions are presented in feet. However, fencing costs are calculated per yard. Since 1 yard equals 3 feet, we need to convert the given dimensions from feet to yards. The formula for conversion is simply dividing the number of feet by 3.

• The length of the field: 70 feet ÷ 3 = 23.33 yards
• The width of the field: 30 feet ÷ 3 = 10 yards

Performing these conversions ensures that all our measurements align, allowing us for accurate calculation of the perimeter in yards.

Cost Calculation

To find the cost of enclosing the field, we must first calculate the perimeter and then apply the cost per yard. Each unit of measurement now should be consistent, which is why converting feet to yards is the first step before calculating the cost.

• First, we multiply the perimeter by the cost per yard.
• If it costs $8 per yard to install a fence, simply multiply the total yardage by this rate.

In this example, the perimeter is approximately 66.66 yards. Thus, the cost to fence the field is 66.66 yards multiplied by $8 per yard, resulting in a total cost of $533.28. By thoroughly following these calculations, we ensure that we budget correctly for any projects.

Perimeter Calculation

The perimeter of a rectangle is the total length around it. For a rectangular field, this involves adding the lengths of all four sides. The formula for the perimeter of a rectangle is:

• twice the sum of the length and the width

In the case of our field, converted to yards, this is 23.33 yards of length plus 10 yards of width. The calculation is:

• Perimeter = 2 * (Length + Width) = 2 * (23.33 + 10) yards = 66.66 yards

This value is essential, as it tells us how much fencing material is needed. Whether building a physical fence or calculating costs, understanding how to determine the perimeter is a necessary geometrical skill.