Question

minimizing heating and cooling costs a building in the shape of a rectangular box is to have a volume of 15,972 ft3 (see the figure). it is estimated that the annual heating and cooling costs will be $2/ft2 for the top, $4/ft2 for the front and back, and $3/ft2 for the sides. find the dimensions of the building that will result in a minimal annual heating and cooling cost. what is the minimal annual heating and cooling cost?

Answer 1

To minimize the annual heating and cooling costs of the building, we need to find the dimensions that will result in a minimal cost. This can be done by solving the volume equation and expressing one dimension in terms of the other. The minimal annual heating and cooling cost can be calculated using the minimized cost function.

Step-by-step explanation:

To minimize the annual heating and cooling costs of the building, we need to find the dimensions that will result in a minimal cost. Let's represent the dimensions of the rectangular box as length (L), width (W), and height (H). Since the volume of the box is given as 15,972 ft³, we have the equation L × W × H = 15,972. To find the minimal cost, we need to minimize the total surface area of the building. The total surface area is given by 2(L × W + L × H + W × H). Using the given costs per square foot, the cost function for the surface area is C = 2(2 × L × H + 4 × L × W + 3 × W × H). We can use the volume equation to express L or W in terms of H, substitute it into the cost function, and then find the minimum by taking the derivative and setting it to zero. By solving this equation, we can find the dimensions that minimize the cost. The minimal annual heating and cooling cost can then be calculated using the minimized cost function.