1) A family is building a patio onto the back of their house with a rectangular area in the center and equal square sections on both sides. The entire length of the patio can be no longer than 20 feet. Let p represent the width of each square section.
assuming a nonzero interior rectangle,
2p < 20
To visualize the patio, let's break it down into its components:
- The rectangular area in the center has a length of p and a width of 20 - 2p, since the total length of the patio is 20 feet and we have two square sections of width p on both sides.
- The area of the rectangular section is simply the product of its length and width, which is p * (20 - 2p), or 20p - 2p^2.
- The square sections on both sides have an area of p * p, or p^2.
- The total area of the patio is then the sum of the rectangular area in the center and the areas of the square sections on both sides, which is (20p - 2p^2) + 2p^2.
To find the maximum area, we can take the derivative of the total area equation with respect to p, set it equal to zero, and solve for p. This will give us the value of p that maximizes the area.
- Let's differentiate the equation with respect to p:
d/dp [(20p - 2p^2) + 2p^2] = 20 - 4p
- Setting this equal to zero:
20 - 4p = 0
- Solving for p:
4p = 20
p = 20/4
p = 5
Therefore, the width of each square section should be 5 feet in order to maximize the area of the patio.