CALCULUS HELP!!!! OPTIMIZATON!!

Question

If 10,800 cm² of material is available to make a box with a square base and an open top, find the largest possible volume (in cm³) of the box.

we did not learn this in class so i am really confused.

Answer 1

Since we are given a surface area and we need to find the volume, the first thing we need to do is find equations for them both.

The box has a square base with length and width x and 4 walls of length x and height h giving:

SA = x² + 4xh = 10800 cm²

Volume of a box like this is just multiplying length, width and height together but since the length = width we get:

V = x²h

From here we need to solve for h (it's easier to get by itself) using the surface are equation (we know what that one is):

h = 10800 - x²

4x

Now we can start solving for the dimensions of the box by plugging in h and taking the derivative of V:

V = x² (10800 - x²) = 10800x - x³

4x 4

V^' = 10800 - 3x²

4

To get the maximum, we set the derivative equal to 0 (this gives us the high point of the parabola) and solve for x:

0 = 10800 - 3x²

4

x = 60 cm

( we can check that this is indeed a maximum by evaluating the second derivative at x = 60 cm and getting a negative number which we do)

Plugging this back into the equation for h we get h = 30cm and solving for V:

V = (60 cm)² (30) = 108000 cm³