Problem 9 - Precalculus Student Solutions Manual 5th

Question

Solve each problem. Perimeter of a Rectangle The perimeter of a rectangle is \(294 \mathrm{cm}\). The width is \(57 \mathrm{cm} .\) Find the length.

Step-by-Step Solution

Answer

The length is 90 cm.

Step 1: Understand the Perimeter Formula

The perimeter of a rectangle is calculated using the formula: \[ P = 2(L + W) \]where \(P\) is the perimeter, \(L\) is the length, and \(W\) is the width.

Step 2: Substitute Known Values

Given the perimeter \( P = 294 \) cm and the width \( W = 57 \) cm, substitute these values into the formula:\[ 294 = 2(L + 57) \]

Step 3: Simplify the Equation

Divide both sides by 2 to simplify:\[ 147 = L + 57 \]

Step 4: Solve for the Length

Subtract 57 from both sides to solve for \(L\):\[ 147 - 57 = L \]\[ L = 90 \]

Step 5: Verify the Solution

Check the solution by substituting \(L = 90\) and \(W = 57\) back into the original perimeter formula:\[ P = 2(90 + 57) = 2(147) = 294 \]The solution is verified as correct.

Key Concepts

Rectangle Perimeter Formula

The perimeter of a rectangle is the total distance around the edge of the rectangle. You can find it using the formula: equation \( P = 2(L + W) \) where

• \( P \) is the perimeter,
  
• \( L \) is the length of the rectangle, and
  
• \( W \) is the width of the rectangle.
  

In our example, the perimeter (\( P \)) is given as 294 cm, and the width (\( W \)) is given as 57 cm. We need to find the length (\( L \)).The formula tells us to double the sum of the length and the width to find the perimeter.

Solving Equations

To find the unknown length (\( L \)), we need to solve the equation given in the problem. We start by substituting the known values into the formula:\[ 294 = 2(L + 57) \]Next, we simplify the equation. First, let's divide both sides by 2:\[ 147 = L + 57 \]Now, we need to isolate \( L \). To do this, subtract 57 from both sides:\[ L = 147 - 57 \]\[ L = 90 \]So, the length of the rectangle is 90 cm. Solving equations often involves these kinds of steps: substitution, simplification, and isolation of the variable.

Verification of Solutions

Once we have a solution, it's important to verify it to make sure it is correct. We do this by plugging the solution back into the original formula and checking our work. In this case, we substitute \( L = 90 \) cm and \( W = 57 \) cm back into the perimeter formula: \[ P = 2(90 + 57) \]\[ P = 2(147) \]\[ P = 294 \]The calculated perimeter matches the given perimeter of 294 cm. This confirms that our solution is correct. Verification is an important step because it ensures that our calculations were done correctly and that our solution makes sense within the context of the problem.