Final answer:
The dimensions (in cm) that will minimize the cost for the box are length x = 5.00 cm, width y = 5.00 cm, and height z = 2.80 cm.
Step-by-step explanation:
To minimize the cost of the box, given that its volume must be 70 cubic centimeters and the cost of the top/bottom is $0.20 per square centimeter and the sides cost $0.10 per square centimeter, we start by expressing the total cost function C(x, y, z), where x, y, and z are the dimensions, in terms of x and y. The cost function C can be written as: C(x, y, z) = 4xy + 2xz. Since V = xyz = 70 cm³ is the volume, we solve for z in terms of x and y to substitute it into C(x, y, z). After substituting, we obtain a function of C in terms of two variables x and y.
Next, by finding the partial derivatives dC/dx and dC/dy and setting them equal to zero, we can locate the critical points. Solving the system of equations for x and y, we find that x = 5.00 cm and y = 5.00 cm. Finally, substituting these values into z = V/(xy), we get z = 2.80 cm, which fulfills the volume requirement. Thus, the dimensions x = 5.00 cm, y = 5.00 cm, and z = 2.80 cm minimize the cost of the box while maintaining the given volume constraint.