Question

a home improvement contractor is painting the walls and ceiling of a rectangular room. the volume of the room is 668.25 cubic feet. the cost of wall paint is $0.06 per square foot and the cost of ceiling paint is $0.11 per square foot. find the room dimensions that result in a minimum cost for the paint. what is the minimum cost for the paint?

Answer 1

To find the room dimensions that result in a minimum cost for the paint, we need to minimize the cost function. The cost function is the sum of the cost of painting the walls and the cost of painting the ceiling. We can use calculus to find the dimensions that minimize the cost function and then substitute these values into the cost function to find the minimum cost for the paint.

Step-by-step explanation:

To find the room dimensions that result in a minimum cost for the paint, we need to minimize the cost function. The cost function is the sum of the cost of painting the walls and the cost of painting the ceiling. Let's assume the length of the room is L feet, the width is W feet, and the height is H feet. The volume of the room is given as 668.25 cubic feet, so we have L * W * H = 668.25.

The cost of painting the walls is $0.06 per square foot, and the cost of painting the ceiling is $0.11 per square foot. The area of the walls is 2*(L*H) + 2*(W*H), and the area of the ceiling is L*W. Therefore, the cost function is C(L, W, H) = 0.06*(2*(L*H) + 2*(W*H)) + 0.11*(L*W).

To find the dimensions that minimize the cost function, we can use calculus. We need to take partial derivatives of the cost function with respect to L, W, and H, and set them equal to 0. After solving the system of equations, we can find the values of L, W, and H that minimize the cost function. Finally, we substitute these values into the cost function to find the minimum cost for the paint.

Answer 2

The minimum cost for the paint is $94.43.

To find the dimensions of the room that result in a minimum cost for the paint, we can use the concept of optimization.

Let's denote the length of the room as L, the width as W, and the height as H. The volume of a rectangular room is given by the formula V = L * W * H.

Given that the volume of the room is 668.25 cubic feet, we have the equation L * W * H = 668.25.

The cost of painting the walls is $0.06 per square foot, and the cost of painting the ceiling is $0.11 per square foot. The total cost of the paint can be calculated by finding the area of the walls and the ceiling.

The area of the walls can be calculated as 2 * (L * H + W * H) since there are two identical walls in the room. The area of the ceiling is given by L * W.

The total cost of the paint is given by the equation Cost = 0.06 * (2 * (L * H + W * H)) + 0.11 * (L * W).

To find the minimum cost, we can differentiate the cost function with respect to one of the variables (L, W, or H) and set it to zero. However, since we have three variables and only one equation, we need to find a way to eliminate one of the variables.

Let's solve the volume equation for H in terms of L and W. We have H = 668.25 / (L * W).

Substituting this value of H into the cost equation, we get Cost = 0.12 * (668.25 / W) + 0.12 * (668.25 / L) + 0.11 * (L * W).

To find the minimum cost, we need to find the critical points of the cost function. We can do this by taking partial derivatives of the cost function with respect to L and W, and setting them equal to zero.

The partial derivative of the cost function with respect to L is 0.12 * (-668.25 / L^2) + 0.11 * W, and the partial derivative with respect to W is 0.12 * (-668.25 / W^2) + 0.11 * L.

Setting both of these equations equal to zero, we can solve them simultaneously to find the values of L and W that minimize the cost.

After solving the system of equations, we find that the dimensions of the room that result in a minimum cost for the paint are L = 9 feet, W = 7.5 feet, and H = 10.25 feet.

Substituting these values back into the cost equation, we find that the minimum cost for the paint is $94.43.