Arectangular storage container with an open top is to have a volume of 10 m3. the length of this base is twice the width. material for the base costs $5 per square meter. material for the sides costs $3 per square meter. find the cost of materials for the cheapest such container. (round your answer to the nearest cent.)

We have to find the cost of materials for the cheapest container. It has a volume of 10 m^3. V = 10 m^3; V = L * W * H; where L is length, W is width and H is height. We know that L = 2 W; 10 = 2 W^2 * H ; H = 10 / 2W^2 ; H = 5 / W^2. Base Area = L * W = 2 W * W = 2 W^2. Side Area = 5 / W^2 * ( 2 W + 4 W ) = 5 / W^2 * 6 W = 30 / W. The Cost: C ( W ) = 5 * 2 W^2 + 3 * 30 / W = 10 W^2 + 90 / W. We have to find the derivative of the cost. C`(W) = 20 W - 90 / W^2 = ( 20 W^3 - 90 ) / W^2. Then: C` ( W ) = 0; 20 W^3 - 90 = 0; 20 W^3 = 90; W^3 = 4.5 ; W = 1.651. Finally the cost for the cheapest container is: C ( 1.651 ) = 10 * ( 1.651 )^2 + 90 / 1.651 = 10 * 2.72568 + 54.5157 = 27.2568 + 54.5157 = 81.7725 ( $81.77 to the nearest cent ). The cost of materials for the cheapest such container is $81.77.

I’m not absolutely sure but i think it’s 7x6x5x4x3x2=5040 because as the first character you can take any of the 7 given then you are left with 5 then 6 then 4 etc.