This is problem 16 Section 4.6 page 246.
A closed box with square base is to be built to house an ant colony. The bottom of the box and all four sides are to be made of material costing dollar/sq ft, and the top is to be constructed of glass costing dollar/sq ft. What are the dimensions of the box of greatest volume that can be constructed for dollars?
NOTE: Let denote the length of the side of the base and denote the height of the box.
some actual numbers would make this easier to answer. . . Not everyone has access to your text -- Or even knows which text that is . . .
To find the dimensions of the box of greatest volume, we need to express the volume of the box in terms of the variables given.
Let's start by determining the dimensions of the base of the box. Since the base is square, its sides will have the same length. Let's denote this length as . So, the length and width of the base would both be .
Next, we need to determine the height of the box, which we'll denote as .
The total surface area of the box can be calculated by adding the area of the base, which is , to four times the area of each of the sides, which are all squares with side length . Thus, the surface area of the box is:
()
The total cost of the box can be calculated by multiplying the total surface area by the cost per square foot, . Therefore, the cost of the box is:
()
Finally, we need to express the volume of the box in terms of the given variables. The volume of a rectangular prism is equal to the product of the length, width, and height. So, the volume of the box is:
()
Now, we can express the volume of the box as a function of the variable . Let's call this function . Thus, we have:
()
To find the dimensions of the box that maximize its volume, we need to find the value of that maximizes the function . This can be done by finding the critical points of , which occur when its derivative is equal to zero.