Problem 89 - Elementary Algebra

Question

A homeowner is comparing the price of putting a fence around the pool area in his backyard and the price of putting a fence around the entire backyard. The pool area is a rectangle that is \(30 \mathrm{ft}\) wide and \(45 \mathrm{ft}\) long. The backyard is a rectangle that is \(80 \mathrm{ft}\) wide and \(100 \mathrm{ft}\) long. The average price of fencing and gates is \(\$ 10.50\) per foot. Find the difference in price to fence the two areas.

Step-by-Step Solution

Answer

The difference in price to fence the two areas is \$2205.

Step 1 - Determine the Perimeter of the Pool Area

The pool area is a rectangle with a width of 30 feet and a length of 45 feet. To find the perimeter of a rectangle, use the formula: \[ P = 2(l + w) \] where \( l \) is the length and \( w \) is the width.Substituting the given values: \[ P_{\text{pool}} = 2(45 + 30) = 2(75) = 150 \text{ feet} \]

Step 2 - Determine the Perimeter of the Backyard

The backyard is a rectangle with a width of 80 feet and a length of 100 feet. Again, use the formula for the perimeter of a rectangle: \[ P = 2(l + w) \] Substituting the given values: \[ P_{\text{backyard}} = 2(100 + 80) = 2(180) = 360 \text{ feet} \]

Step 3 - Calculate the Cost to Fence the Pool Area

The cost to fence an area is calculated by multiplying the perimeter of the area by the cost per foot of fencing. The perimeter of the pool area is 150 feet and the cost per foot is \$10.50:\[ \text{Cost}_{\text{pool}} = 150 \times 10.50 = 1575 \text{ dollars} \]

Step 4 - Calculate the Cost to Fence the Backyard

Using the same method as in the previous step, multiply the perimeter of the backyard by the cost per foot of fencing. The perimeter of the backyard is 360 feet and the cost per foot is \$10.50:\[ \text{Cost}_{\text{backyard}} = 360 \times 10.50 = 3780 \text{ dollars} \]

Step 5 - Find the Difference in Cost

Subtract the cost to fence the pool area from the cost to fence the backyard to find the difference in price:\[ \text{Difference} = 3780 - 1575 = 2205 \text{ dollars} \]

Key Concepts

Geometry

Geometry helps us understand shapes and their properties. In this exercise, the shapes we are working with are rectangles. Understanding the basics of perimeter calculation is crucial when dealing with enclosed areas like the pool and the backyard.
The key formula used here is: \[ P = 2(l + w) \]
This formula works because a rectangle has two sets of identical sides. By adding the length (\( l \)) and the width (\( w \)) and then multiplying by 2, we account for all four sides of the rectangle. Here are the steps:
* For the pool area: Length is 45 feet and width is 30 feet
* For the backyard: Length is 100 feet and width is 80 feet

Knowing how to calculate perimeter helps solve real-life problems, like fencing an area, whether it’s for security or simply marking boundaries.

Algebra

Algebra helps us work with variables and equations. In this solution, we use algebra to systematically solve for the perimeter and then the cost. The formula for the perimeter of a rectangle, \[ P = 2(l + w) \], is an algebraic equation.
After you substitute the values into the equation, it becomes a straightforward arithmetic problem:
* For the pool area: \[ P_{\text{pool}} = 2(45 + 30) = 2(75) = 150 \text{ feet} \]
* For the backyard: \[ P_{\text{backyard}} = 2(100 + 80) = 2(180) = 360 \text{ feet} \]
Once you have the perimeters, you move on to costing. Multiplication here is another basic algebraic operation. It allows us to find the total cost of fencing by multiplying the perimeter by the cost per foot:
* For the pool: \[ \text{Cost}_{\text{pool}} = 150 \times 10.50 = 1575 \text{ dollars} \]
* For the backyard: \[ \text{Cost}_{\text{backyard}} = 360 \times 10.50 = 3780 \text{ dollars} \] Understanding these basic algebraic computations helps students see the practical application of algebra in everyday life.

Cost Calculation

Cost calculation is a crucial part of budgeting and planning for any project. In this exercise, we determine the cost of fencing by first finding the perimeters and then multiplying by the cost per foot. Here’s how it’s done step-by-step:
1. Find the perimeter of the area to be fenced
2. Multiply the perimeter by the cost per foot of fencing material (given as $10.50 per foot)

Again, for the pool area, the cost calculation is:
\[ \text{Cost}_{\text{pool}} = 150 \times 10.50 = 1575 \text{ dollars} \] For the backyard, it’s:
\[ \text{Cost}_{\text{backyard}} = 360 \times 10.50 = 3780 \text{ dollars} \]
Lastly, to find the difference in cost between the two areas, subtract the cost of fencing the pool from the cost of fencing the backyard:
\[ \text{Difference} = 3780 - 1575 = 2205 \text{ dollars} \]
This step ensures we understand not just the calculation but also the implications of different sized projects on cost.

Rectangles

Rectangles are a fundamental shape in geometry. They have opposite sides that are equal in length and four right angles. The properties of rectangles make calculating perimeters straightforward using the formula: \[ P = 2(l + w) \]
In practical applications, rectangles can be seen in everyday objects, land plots, and especially man-made structures like swimming pools and backyards.

• For the pool: 30 feet wide and 45 feet long
• For the backyard: 80 feet wide and 100 feet long

Always double-check your dimensions to ensure accuracy, especially when these measurements affect costs in real-world applications. By learning to handle basic rectangular shapes, students can prepare for more complex geometric forms and calculations in the future.