A square is cut from each corner of a sheet of metal with dimensions 12cm by 10cm to form an open rectangular container find using calculus the max volume of the container?

Answer 1

If you cut a square of length x cm from each corner of the sheet, you will be able to fold the flaps your form up to create a box that is 12-2x long, 10-2x wide, and x high. Using the formula for the volume of a rectangular prism (length*width*height) we find the volume as a function of x to be

Volume(x)=(12-2x)*(10-2x)*x=4x3-44x²+120x

In order to maximize this volume, we need to find where its derivative is 0.

d/dx(Volume(x))=12x²-88x+120=0

Since this quadratic cannot be factored, we will have to solve it using the quadratic formula, giving us that

x=1/3 * (11±√31)~1.81075, 5.52259

While we may have two answers, only one of them is valid. Since we need to cut this amount out of a 10 by 12 box, we can't cut 5.52259 from each corner or else we would be cutting more than 10 inches (since you cut twice per side) on one side, which is more than whats there to cut! Therefore, the only possible solution is the 1.81075. To make sure this is a max and not a min though, we need to make sure the second derivative is negative there.

d²/dx²(Volume)=12x-88=6*(1.81075)-44=-44.542

Since it was negative like we wanted, we know that the maximum volume is achieved when we cut out 3.62149 inches. Plugging that value of x into our original equation for volume we get

Volume(1.81075)=1.810753-22*1.81075²+120*1.81075=96.7706 cubic centimeters.