Question

Question content area top part 1 a storage shed is to be built in the shape of a box with a square base. it is to have a volume of 735 cubic feet. the concrete for the base costs ​$6 per square​ foot, the material for the roof costs ​$9 per square​ foot, and the material for the sides costs ​$3.50 per square foot. find the dimensions of the most economical shed.

Answer 1

To find the most economical dimensions of a storage shed with a square base and a volume of 735 cubic feet, we need to minimize the total cost using the cost per material for each part and the total volume. The dimensions that minimize the cost can be found using calculus by setting the derivative of the cost equation to zero.

Step-by-step explanation:

The question is asking us to minimize the cost of building a storage shed with a volume of 735 cubic feet. The shed has a square base, and different materials are used for different parts of the shed, each with its own cost per square foot: $6 for the concrete base, $9 for the roof, and $3.50 for the sides. To find the most economical dimensions of the shed, we need to minimize the total cost.

Let x represent the length of one side of the square base, and h represent the height of the shed. The volume of the shed, which is 735 cubic feet, is given by the formula V = x²h. The total cost C consists of the cost of the base, the roof, and the sides. The cost equation is: C = (base cost) + (roof cost) + (sides cost) = 6x² + 9x² + 4(3.50)xh. By substituting h = 735/x² into the cost equation and taking the derivative of C with respect to x, we can set the derivative equal to zero to find the critical points, and then determine which gives the minimum cost for the shed. This calculation would involve calculus to solve for the dimensions that minimize the cost.