Question

"You are going to make a trash bin with a square base , vertical sides and no top. If you have 108 ft2 of material available, determine the dimensions of the box with a maximum volume? You can neglect any materials wasted during construction."

Answer 1

Dimension of trash bin is 6 ft x 6 ft x 3ft .

Step-by-step explanation:

Let h be the height of box and a be the side of base square.

We have 108 ft² material.

Surface area of trash bin = 108 ft²

Surface area of trash bin = a² + 4 ah

a² + 4 ah = 108

       4ah = 108 - a²

            $h=\frac{108-a^2}{4a}$

Volume of trash bin, V = a²h

           $V=a^2\times \frac{108-a^2}{4a}=27a-0.25a^3$

At maximum volume differentiation of volume will be zero

               $dV=27da-0.25\times 3a^2da\\\\0=27da-0.75a^2da\\\\a^2=36\\\\a=6ft$

Surface area of trash bin = a² + 4 ah = 108

                                              36 + 4 x 6 x h = 108

                                                           h = 3 ft

Dimension of trash bin is 6 ft x 6 ft x 3ft .