13. The perimeter of a rectangle is 80 feet. Find the dimensions if the length is 5 feet longer than four times the width. Then find the area of the rectangle.

To solve for the dimensions of the rectangle, we start by setting up the equations based on the information given. Let the width be \( w \) feet. According to the problem, the length \( l \) can be expressed as \( l = 4w + 5 \). The perimeter \( P \) of a rectangle is given by the formula \( P = 2l + 2w \). Since the perimeter is 80 feet, we can write the equation \( 2(4w + 5) + 2w = 80 \). Now, simplify and solve for \( w \): \[ 2(4w + 5) + 2w = 80 \] \[ 8w + 10 + 2w = 80 \] \[ 10w + 10 = 80 \] \[ 10w = 70 \] \[ w = 7 \] Now that we have the width, we can find the length: \[ l = 4(7) + 5 = 28 + 5 = 33 \] So, the dimensions are 33 feet (length) and 7 feet (width). The area \( A \) of the rectangle can be calculated with the formula \( A = l \times w = 33 \times 7 = 231 \) square feet. The dimensions of the rectangle are 33 feet (length) by 7 feet (width), and the area is 231 square feet. Now let’s jazz it up with some context: Who knew rectangles could be such role models in geometry? Back in ancient times, they were key players in land surveying and architecture—think ancient Egypt, where the pyramids were crafted with precision and rectangular plots. This mathematical concept helped civilizations build structures that stood the test of time! And speaking of practical uses, knowing how to calculate the area of a rectangle comes in handy every day—whether you are designing a garden, laying down flooring, or even plotting a new room layout. Being comfortable with this kind of math can transform your living space and help budget materials accurately. So roll up your sleeves, grab that tape measure, and get calculating!