Please include detailed steps to the solution. Will rate thumbs up for good legi

Question

Please include detailed steps to the solution. Will rate thumbs up for good legibile answer. Thanks.

Joseph wishes to make an closed-top box using the least amount of materials possible. The base of this box has a length which is twio times its width and the volume is to be 72 cm3 (a) Express the amount of material needed (surface area) as a function of one variable and give the domain of the function. (b) Find the dimensions needed so that Joseph uses the least amount of material. Verify that your result gives the minimunm surface area.

Explanation / Answer

Closed top box
Base length = 2x
Base width = x
Ht = y

We know 2x*x*y = 72
So, x^2y = 36

Amt of material needed is :
2LH + 2BH +2LB
2(2x)(y)+ 2(x)(y) + 2(2x)(x)

A = 4xy + 2xy + 4x^2

A = 4x^2 + 6xy

Now, plug in y = 36/x^2 :

A = 4x^2 + 6x(36/x^2)

A = 4x^2 + 216/x ----> ANS

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b)
Now for least amt of material,
dA/dx = 0 :

dA/dx = 8x - 216/x^2 = 0

8x = 216/x^2

x^3 = 27

So, x =3
Thus, length = 2x = 6

And ht = 36/x^2 ---> 36/9 ---> 4

So, dimensions are :
LEngth = 6 cm
Width of base = 3 cm
Ht of box = 4 cmc

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