Problem 10 - Precalculus Student Solutions Manual 5th

Question

Solve each problem. Perimeter of a Storage Shed Michael Gomski must build a rectangular storage shed. He wants the length to be 6 ft greater than the width, and the perimeter will be \(44 \mathrm{ft}\). Find the length and the width of the shed.

Step-by-Step Solution

Answer

Width: 8 ft, Length: 14 ft

Step 1 - Define Variables

Let the width of the shed be denoted as \(w\) (in feet). Then the length must be \(w + 6\) (in feet) according to the problem statement.

Step 2 - Set Up the Perimeter Equation

The perimeter of a rectangle is given by the formula \(P = 2L + 2W\). Here, \(L\) is the length and \(W\) is the width. Substitute the given perimeter (44 ft) into the formula: \[ 44 = 2(w + 6) + 2w \]

Step 3 - Simplify the Equation

Expand and combine like terms: \[ 44 = 2w + 12 + 2w \] Simplify this equation: \[ 44 = 4w + 12 \]

Step 4 - Solve for Width

Isolate the variable \(w\): \[ 44 - 12 = 4w \] \[ 32 = 4w \] Divide by 4: \[ w = 8 \]

Step 5 - Find the Length

Substitute the width back into the equation for the length: \[ L = w + 6 \] \[ L = 8 + 6 \] \[ L = 14 \]

Key Concepts

Perimeter of a Rectangle

In geometry, the perimeter of a rectangle is the total distance around the shape. To find the perimeter, we add up the lengths of all four sides. Since a rectangle has two pairs of equal sides, this simplifies to the formula: \(P = 2L + 2W\).
Here:

• \(P\) represents the perimeter, which is the total length around the rectangle.
• \(L\) stands for the length of the rectangle, which is one of the longer sides.
• \(W\) stands for the width, which is one of the shorter sides.

By knowing two of these three values, we can always find the third. In our problem, we know the perimeter (44 feet) and we want to find the length and width of Michael Gomski's storage shed. This brings us to the importance of defining variables.

Variable Definition

To solve any algebra problem, clearly defining your variables is crucial. Variables are symbols (usually letters) that stand for unknown values.
In our problem:

• We let \(w\) represent the width of the shed.
• Since the length is 6 feet greater than the width, we represent the length as \(w + 6\).

Using variables makes it easier to set up equations based on the problem’s conditions. This leads us to the next step: setting up and solving the equation.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. Let’s look at how we solve for the width:
We start with the perimeter equation: \(44 = 2(w + 6) + 2w\).
Here's the step-by-step breakdown:

• First, distribute the 2 to both terms inside the parenthesis: \(2(w + 6)\) becomes \(2w + 12\).

Now the equation looks like this: \(44 = 2w + 12 + 2w\).
Combine like terms: \(44 = 4w + 12\).
To isolate \(w\), subtract 12 from both sides: \(44 - 12 = 4w\) gives us \(32 = 4w\).
Finally, divide by 4: \(w = 8\). This gives us the width.
To find the length, plug \(w\) back into the equation for the length: \(L = w + 6 = 8 + 6 = 14\).
So, the width is 8 feet and the length is 14 feet. Linear equations are very useful for solving real-world geometry problems like these.