Question

A museum is building a special box to display a new exhibit. The box needs to have a volume of 20 feet and a height of 2 feet. The front of the box, which is shaded in the diagram below, will be made of glass, which costs 4 per square foot. The top, sides, back, and bottom of the box will be made of a metal that costs1 per square foot. Let a and b be the length and width of the box, as shown below. Find values of a and b which minimize the cost of the box. What will the cost be in this case?

Answer 1

To find the values of a and b that minimize the cost of the box, we set up a cost function and take the derivative with respect to a and b. Solving these equations, we find that a = 4/5 and b = 4/5. Plugging these values back into the cost function, the cost of the box is $18.80.

Step-by-step explanation:

To find the values of a and b that minimize the cost of the box, we need to minimize the cost function. Let's define the cost function as C = 4ab + 2(a+b)(20-2) + ab, where the first term represents the cost of the glass front, the second term represents the cost of the metal sides, back, and bottom, and the third term represents the cost of the top. We can simplify this function to C = 4ab + 72 - 4(a+b) + ab. To minimize this function, we can take the derivative with respect to both a and b, set them equal to zero, and solve for a and b.

Taking the derivative with respect to a, we get dC/da = 4b + b - 4 = 5b - 4. Setting this equal to zero, we get 5b - 4 = 0.

Taking the derivative with respect to b, we get dC/db = 4a + a - 4 = 5a - 4. Setting this equal to zero, we get 5a - 4 = 0.

Solving these two equations simultaneously, we find that a = 4/5 and b = 4/5. Plugging these values back into the cost function, we get C = 4(4/5)(4/5) + 72 - 4(4/5+4/5) + (4/5)(4/5) = 64/5 + 72 - 32/5 + 16/25 = 470/25 = 18.8. Therefore, the cost of the box in this case is $18.80.