The amount of paint needed to cover the calls of a room varies jointly as the perimeter of the room and the height of the wall. If a room with a perimeter of 60 feet and 8-foot walls requires 4.8 quarts of paint, find the amount of paint needed to cover the walls of a room with a perimeter of 65 feet and 10-foot walls.

Wall Area room 1 = 60 * 8

Wall Area room 2 = 65 * 10
so
4.8 ( 65*10 / 60*8)

A rectangular box with volume 490 cubic feet is built with a square base and top. The cost if $1.50 per square foot for the top and bottom and $2.00 per square foot for the sides. Let x represent the length of a side of the base. Express the cost of the box as a function of x.

height = 490/x^2

cost = 1.50*2x^2+ 2.00*490/x^2

suppose that the manufacturer of a gas clothes dryer has found that, when the unit price is p dollars, the revenue R(in dollars) is R(p)= -9p^2 +18,000p

What unit price should be established for the dryer to maximize revenue? What is the maximum revenue?

To solve this problem, we need to find the constant of variation that relates the amount of paint needed to the perimeter and height of the walls. Once we have this constant, we can use it to calculate the amount of paint needed for different room dimensions.

Let's denote the amount of paint needed as P, the perimeter as P, and the height of the walls as H.
According to the problem, the amount of paint needed (P) varies jointly as the perimeter (P) and the height of the walls (H). This can be expressed as:

P = k * P * H

where k is the constant of variation.

To find k, we can use the given information from the first scenario: a room with a perimeter of 60 feet, 8-foot walls, and requiring 4.8 quarts of paint. Substituting these values into the equation, we get:

4.8 = k * 60 * 8

Simplifying this equation, we find:

k = 4.8 / (60 * 8)
k = 0.01

Now that we have the value of k, we can calculate the amount of paint needed for the second scenario, where the perimeter is 65 feet and the height of the walls is 10 feet. Using the equation again, we have:

P = 0.01 * 65 * 10

Simplifying this equation, we find:

P = 6.5 quarts of paint

Therefore, the amount of paint needed to cover the walls of a room with a perimeter of 65 feet and 10-foot walls is 6.5 quarts.