Problem 57 - Algebra and Trigonometry

Question

Find the length and width of a rectangle whose perimeter is 36 feet and whose area is 77 square feet.

Step-by-Step Solution

Answer

The dimensions of the rectangle are \(L = 11\) feet and \(W = 7\) feet.

Step 1: Use the Given Perimeter to Form an Equation

From the given equation, we can derive an equation relating the length \(L\) and width \(W\) of the rectangle from the perimeter formula. The formula to calculate the perimeter of a rectangle is \(2L + 2W\). Replacing 36 feet, this becomes: \(2L + 2W = 36\). Re-arranging this equation, we have: \(L = 18 - W\).

Step 2: Use the Given Area to Form a Second Equation

Similarly using the formula for the area of a rectangle which is \(L \times W\), and substituting the given 77 square feet area, we can form another equation: \(L \times W = 77\). Now substitute the value of \(L\) from first step into this equation, we have: \((18 - W) \times W = 77\).

Step 3: Solve the Quadratic Equation

Rearranging the equation \((18 - W) \times W = 77\) gives us a quadratic equation: \(W^2 - 18W + 77 = 0\). By solving this equation, we can find the possible values for the width \(W\).

Step 4: Solve for Length

Once we have the width \(W\), we can substitute this into the equation from Step 1, \(L = 18 - W\), to find the value for length \(L\).