Problem 278 - Elementary Algebra

Question

Jose just removed the children's playset from his back yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a 50 foot roll of fence in his garage that he plans to use. To fit in the backyard, the width of the garden must be 10 feet. How long can he make the other length?

Step-by-Step Solution

Answer

15 feet

Step 1: Identify the Problem

The problem requires finding the length of a rectangular garden given its width and the total perimeter of the garden.

Step 2: Understand the Perimeter Formula

The perimeter of a rectangle can be calculated with the formula: \( P = 2L + 2W \) where \( L \) is the length and \( W \) is the width.

Step 3: Substitute Known Values

In this problem, the total perimeter \( P \) is 50 feet and the width \( W \) is 10 feet. Substitute these values into the formula: \( 50 = 2L + 2(10) \).

Step 4: Simplify the Equation

Simplify the equation step-by-step: \( 50 = 2L + 20 \).

Step 5: Isolate the Variable

To solve for \( L \) (the unknown length), first subtract 20 from both sides of the equation: \( 50 - 20 = 2L \) which simplifies to: \( 30 = 2L \).

Step 6: Solve for Length

Divide both sides by 2 to find \( L \): \( \frac{30}{2} = L \), which simplifies to: \( L = 15 \).

Key Concepts

Rectangular Perimeter Calculation

When you're dealing with any rectangle, understanding how to calculate the perimeter is essential. The perimeter is the total distance around the outer edge of the rectangle. To find the perimeter, you use the formula: d
\( P = 2L + 2W \) where \( P \) is the perimeter, \( L \) is the length, and \( W \) is the width. Simply put, multiply the length by 2 and the width by 2, then add those results together. In Jose's case, the garden's width is known (10 feet) and he has 50 feet of fencing, which gives us the perimeter. By plugging these values into the formula, we can solve for the length. It's a straightforward way to ensure you have enough material to enclose a space.

Basic Algebra

When solving problems that involve algebra, it’s like being a detective with numbers. You look for clues and piece them together using a sequence of steps. Here, we use the perimeter formula to find the missing value. After setting up our equation with the known perimeter and width:
\( 50 = 2L + 2(10) \)
we simplify to make it easier to solve. This involves combining like terms and isolating the variable. It's just like cleaning your room in steps so you can find what you’re looking for.Through the simplification:
\( 50 = 2L + 20 \) we subtract 20 from both sides to isolate the term with the variable:
\( 50 - 20 = 2L \) or \( 30 = 2L \).This preparation sets us up perfectly for our next step, which is solving for the length.

Solving for Length

The final step is to solve for the length. Here, we need to find the value of \( L \) that satisfies our equation. From:
\( 30 = 2L \), we divide both sides by 2 to isolate \( L \):
\( \frac{30}{2} = L \), which gives us \( L = 15 \). So, Jose can make the length of his garden 15 feet. By following the steps sequentially—using the perimeter formula, applying basic algebra to simplify, and finally isolating the variable—we can determine the required measurements. Remember, practicing these steps with different problems will sharpen your skills and make these concepts second nature!