Question

you need to construct an open-top rectangular box with a square base that must hold a volume of exactly 425 cm3. the material for the base of the box costs 5 cents/cm2 and the material for the sides of the box costs 5 cents/cm2. find the dimensions for a box that will minimize the cost of the materials used to construct box. please show your answers to at least 4 decimal places.

Answer 1

The dimensions for a box that will minimize the cost of the materials used to construct box length is 8.794 and height is 825.

Given:

In the question:

Rectangular box with a square base that must hold a volume of exactly 425 cm3.

The material for the base of the box costs 5 cents/cm2

and, the material for the sides of the box costs 5 cents/cm2.

To find the dimensions for a box that will minimize the cost of the materials used to construct box.

Now, According to the question:

Volume of box is 425 cm^3

V = 425$cm^3$

Let the dimension of base is L x L and height of box be h

V = $L^2$ x h

Cost of base $c_1$ =  $L^2$ x 5 = 5$L^2$

Cost of sides $c_2$ = (4Lh) x 4

$c_2$ = 16Lh

Total Cost of the material is  = $c_1+c_2$

C =  5$L^2$  + 16Lh

C = 5$L^2$ + 16L x 425/$L^2$

C = 5$L^2$ + $\frac{16 (425)}{L}$

Differentiate C w.r.t L to get maximum/minimum Value

dC/dL = 10L -  $\frac{16 (425)}{L^2}$

L = 8.794

V = $L^2$ x h

425 = (8.794)^2 x h

So,

h = 825

Hence, The dimensions for a box that will minimize the cost of the materials used to construct box length is 8.794 and height is 825.

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