Answer :
The dimensions of the tank with minimum weight is height= 7 and width and length= 14.
The minimum surface area implies that the tank has minimum weight.
Given that the base is square, its width is equal to its length.
Let x= the base width and length of the tank.
Let h = the height .
We are given the volume of the tank that is 1372 ft^3. The formula is given by
V = hx^2
We're given that V = 1,372 ft^3, so
hx^2 = 1,372
h = 1,372 / x^2 ----------(equation 1)
Surface area can be calculated using the below given formula.
A = x^2 + 4xh
Substituting the value of euation 1, h in A, we get,
A = x^2 + 4x(1,372 / x^2)
= x^2 + 5,488 / x
To get the minimum surface area to get the minimum weight, we need to differentiate A with respect to x and equate it to zero.
dA/dx = 2x - 5,488 / x^2 = 0
2x^3 - 5,488 = 0
x =[tex]\sqrt[3]{\frac{5,488}{2} }[/tex]
x= 14
We got the value of x, we substitute the value in equation 1, we get the value of h.
h = 1,372 / 14^2
h= 7
So, the dimensions of the tank with minimum weight are width and length is 14 and height is 7.
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