Problem 80 - College Algebra

Question

The perimeter of a rectangle is 180 feet. Describe the possible lengths of a side if the area of the rectangle is not to exceed 800 square feet.

Step-by-Step Solution

Answer

The possible lengths of a side of the rectangle are between 0 and 80 feet.

Step 1: Express Width in Terms of Length

Given that the perimeter of the rectangle is 180 feet, we can use the formula for perimeter to express width (w) in terms of length (l): \(w = 90 - l\). This comes from simplifying the formula \(2*(l + w) = 180\) .

Step 2: Substitute Width into the Area Formula

Next, substitute \(w = 90 - l\) into the area formula (\(A = l * w\)) to express the area in terms of length. The area becomes \(A = l * (90 - l)\).

Step 3: Find Length Values for Given Area

For the area not to exceed 800 square feet, the equation \(l * (90 - l) ≤ 800\) must satisfy. Solving this equation gives length values between 0 and 80 feet.