A shipping company must design a closed rectangular shipping crate with a square base. The volume is 3072ft3. The material for the top and sides costs $4 per square foot and the material for the bottom costs $2 per square foot. Find the dimensions of the crate that will minimize the total cost of material.

Answer:The dimensions are: length = 16 feet, width = 16 feet and height = 12 feetor Dimensions are  16 ft *16 ft *12 ftStep-by-step explanation:Given: Volume of rectangular with square base is 3072 cubic feet.Let "x" be the side of the square and "h" be the height of the rectangular.Volume of the rectangular (V) = area of the base * heightNow plug in the given value V = 30723072 = [tex]x^2 *h[/tex] -------------------(1)Let's find the area of the top.Which is the square. The area of the top = [tex]x^2[/tex]The area of the bottom is the same = [tex]x^2[/tex]The area of the side = x*hThere are 4 sides, therefore, the areas of the 4 sides = 4xhNow let's find the total cost.Given: The material for the top and sides costs $4 per square foot and the material for the bottom costs $2 per square foot. Total cost (T)= [tex]4x^2 +4*4xh + 2x^2\\= 4x^2 + 16xh + 2x^2\\= 6x^2 + 16xh[/tex]Now let's find h from the equation (1)h = [tex]\frac{3072}{x^2}[/tex]Now plug in h = [tex]\frac{3072}{x^2}[/tex] in Total cost (T), we getT = [tex]6x^2 + 16x(\frac{3072}{x^2} )[/tex]Simplifying the above, we getT = [tex]6x^2 + \frac{49152}{x}[/tex]To find the minimize the total cost, we need to find the derivative of T with respect to x.T'(x) = 12x - [tex]\frac{49152}{x^2}[/tex]Now set the derivative equal to zero and find the value of x. 12x - [tex]\frac{49152}{x^2}[/tex] = 0[tex]\frac{12x^3  - 49152}{x^2} = 0[/tex][tex]12x^3 - 49152 = x^2.0\\12x^3 - 49152 = 0\\12x^3 = 49152\\[/tex]Dividing both sides by 12, we get[tex]x^3 = 4096[/tex]Taking the cube root on both sides, we getx = 16Therefore, the length and the width of the base is 16Now we have to find height.height =  [tex]\frac{3072}{x^2}[/tex]Now plug in x = 16 in the above height, we getheight = [tex]\frac{3072}{16^2} \\= \frac{3072}{256} \\= 12[/tex]Therefore, the height of the rectangular shipping crate is 12 feet.So, the dimensions of the crate that will minimize the total cost of material are length = 16 feet, width = 16 feet and height = 12 feet.