Final answer:
To minimize cost, express the box's dimensions using the width as a single variable, establish a cost function, find its derivative, and solve for the width that sets the derivative to zero. Then calculate the box length and height from this width.
Step-by-step explanation:
To determine the dimensions of the box that will minimize the cost, we must first establish the dimensions in terms of a single variable. Let the width of the base be w inches, which makes the length 6w inches, since the base length is 6 times the base width. Since the volume is 20 in³, the height can be expressed as 20/(6w²). The cost function will be the sum of the cost of the sides and the top/bottom.
The area of the top and bottom (two rectangles each of dimension 6w by w) is 2 × 6w × w = 12w², and the cost is $15/in², resulting in a cost of 180w² dollars. The area of the sides consists of two rectangles of dimension 6w × height and two rectangles of dimension w × height. The surface area for the sides is 2(6w × 20/(6w²)) + 2(w × 20/(6w²)) = 40/w, and the cost is $3/in², making the cost 120/w dollars.
Combining these, we obtain the total cost function: C(w) = 180w² + 120/w. To minimize the cost, we take the derivative of the cost function with respect to w, set it to zero, and solve for w. This will give us the value of w that minimizes the cost. Then, find the length by multiplying w by 6 and the height using the volume equation.
To summarize, set up the total cost equation as a function of width, differentiate, find the minimum, then solve for the other dimensions. That way, we will have the dimensions that minimize the total cost for the material needed for the box.