Find the dimensions that will minimize cost. (Let x represent the length of the sides of the square base and let y represent the height. Round your answers to two decimal places.)

Question

A storage box with a square base must have a volume of 80 cubic centimeters. The top and bottom cost $0.40 per square centimeter and the sides cost $0.20 per square centimeter.

X=___cm

Y=___cm

Answer 1

The box has volume V = x^2 * y as volume is length time width times height. The areas of the top and bottom side of the box both equal x^2 and the areas of the sides of the box all equal x * y. The are 4 side of the box each with cost $ 0.2 per square cm plus the top and bottom each with cost $ 0.4 per square cm, so the total cost C = 4 * 0.2 * x * y + 2 * 0.4 * x ^ 2. We can then plug in for the volume V = 80 and solve for y in term of x. Solving this we get y = 80 / x^2. Plugging this into the cost equation and simplifying, we get C = 64 / x + 0.8 * x^2. We can then minimize the cost function by taking the derivative of C with respect to x and setting it equal to zero. Solving this, we get x = (40)^1/3 and plugging this back into the equation y = 80 / x^2 we get y = 2 * (40)^1/3. To ensure this is a local minimum not a maximum we can take the second derivative of the cost function with respect to x and plug in x = (40)^1/3. Plugging in we get that the second derivative is positive which means the function is concave up and this a minimum. This confirms that we found the dimensions that minimize cost.