Problem 80 - Algebra and Trigonometry

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

Solving the quadratic gives two possible lengths for the rectangle. Using these lengths, the corresponding widths can be calculated. This gives the dimensions of the rectangle that satisfy the conditions of the problem.

Step 1: Express the width in terms of the length from the Perimeter formula.

From the perimeter formula we can express 'width' in terms of 'length'. Perimeter = 2 * (length + width), means that 180 = 2 * (length + width). Solving for 'width' gives \(width = 90 - length\).

Step 2: Substitute the width in the area formula.

Now, substitute \(width = 90 - length\) in \(Area = length * width\) to obtain \(Area = length * (90 - length)\).

Step 3: Solve the inequality given by the area constraint.

The area shouldn't exceed 800 ft². So, solve the inequality \(length * (90 - length) \leq 800\). Multiply terms to get \(90length - length² \leq 800\). Rearrange to get a quadratic equation \(length² - 90length + 800 = 0\). Solve the quadratic to get lengths that make the area ≤ 800 ft².

Step 4: Solve the quadratic equation.

Solving the quadratic \(length² - 90length + 800 = 0\), we get the possible 'length' values. The 'width' can then be calculated using \(width = 90 - length\). Remember that the length and width should be positive numbers.