Problem 16 - Elementary Algebra

Question

For Problems 11-32, use the geometric formulas given in this section to help solve the problems. (Objective 3 ) Find the area of a cement walk 3 feet wide that surrounds a rectangular plot of ground 86 feet long and 42 feet wide.

Step-by-Step Solution

Answer

The area of the cement walk is 804 square feet.

Step 1: Identify the Problem

We need to find the area of a cement walk that surrounds a rectangular plot of ground. The plot is 86 feet long and 42 feet wide, and the walk is 3 feet wide all around.

Step 2: Calculate the Total Dimensions Including the Walk

The walk surrounds the plot, adding 3 feet on each side. Therefore, the total length including the walk is the original length plus two times the width of the walk: \[86 + 2 \times 3 = 92 \text{ feet}\]Similarly, the total width including the walk is:\[42 + 2 \times 3 = 48 \text{ feet}\]

Step 3: Calculate the Area of the Whole Rectangle Including the Walk

Now, compute the area of the entire rectangle (plot + walk) using the dimensions found:\[\text{Area of whole rectangle} = 92 \times 48 = 4416 \text{ square feet}\]

Step 4: Calculate the Area of the Rectangular Plot Inside

Next, calculate the area of just the rectangular plot of ground:\[\text{Area of plot} = 86 \times 42 = 3612 \text{ square feet}\]

Step 5: Find the Area of the Cement Walk

The area of the cement walk is the area of the whole rectangle minus the area of the plot:\[\text{Area of walk} = 4416 - 3612 = 804 \text{ square feet}\]

Key Concepts

Area Calculation

In mathematics, calculating the area is crucial for determining how much space a shape covers. When dealing with surfaces like plots of land or paving walks, knowing how to calculate area helps in estimating materials needed for construction or gardening. To find the area of a simple shape like a rectangle, we multiply its length by its width. This basic formula, \( \text{Area} = \text{length} \times \text{width} \), applies to any rectangle, no matter how large or small.

Possessing a clear understanding of how to calculate area is beneficial in daily life. For example, when renovating your garden or building a driveway, area calculations can guide you in ordering the correct amount of materials. Remember:

• Always use consistent units (e.g., feet, meters) for your calculations.
• Double-check your measurements before proceeding with calculations.
• For complex setups, break them into smaller, manageable parts.

Rectangular Plot

A rectangular plot is a piece of land shaped in the form of a rectangle. Rectangles have four sides, with opposite sides being equal in length. In this exercise, you are dealing with a rectangular plot measuring 86 feet in length and 42 feet in width.

Understanding properties of a rectangle can aid in various calculations. Here are the fundamental characteristics:

• The perimeter of a rectangle is \( 2(\text{length} + \text{width}) \).
• The area is \( \text{length} \times \text{width} \).
• Its diagonals are equal in length.

In practical contexts, knowing your plot’s area helps you understand its capacity and utility for activities such as landscaping, setting up a garden, or building small structures.

Cement Walk

A cement walk is a pathway usually made from concrete that adds both aesthetic appeal and functionality around or within properties. In this scenario, the cement walk surrounds the rectangular plot, adding a uniform width around it.

To find the cement walk area:

• First, determine the external dimensions of the whole area (rectangular plot plus cement walk). Add twice the walk’s width to both the plot’s length and width.
• Calculate the area of the larger rectangle (plot + walk) using these new dimensions.
• Subtract the area of the original plot from this larger area to find the area occupied specifically by the walk.

This subtraction gives you the walk's area. It ensures you are only considering the space covered by the path, which is necessary for planning construction and material requirements efficiently.