A silo (base not included) is to be constructed in the form of a cylinder surmou

Question

A silo (base not included) is to be constructed in the form of a cylinder surmounted by a hemisphere. The cost of construction per square unit of surface area is

8

times as great for the hemisphere as it is for the cylindrical sidewall. Determine the dimensions to be used if the volume is fixed at

2000

cubic units and the cost of construction is to be kept to a minimum. Neglect the thickness of the silo and waste in construction

The radius of the cylindrical base (and of the hemisphere) is blank ft

The height of the cylindrical base is blank ft

Explanation / Answer

solution-:

Let k be the cost of construction for the cylindrical sidewall, so 8k is the corresponding cost for the hemisphere.

Total cost of construction is C = 2k (r^2 - ^2) + 16kr^2.

We need to get rid of one variable. Let's solve the volume equation for r^2:

2000 = ^2 (r^2 - ^2) ==> (r^2 - ^2) = 2000 / ^2

r^2 - ^2 = 16 x 10^6 / ^4 ==> r^2 = ^2 + 16 x 10^6 / ^4

Substituting, C = 2k (2000 / ^2) + 16k (^2 + 16 x 10^6 / ^4)

C = 2k [(2000/)^(-1) + 3^2 + 48 x 10^6^(-4)]

(1 / 2k) dC/d = (-2000/)^(-2) + 16 - 192 x 10^6^(-5) = 0

6^6 - (2000/)^3 - 192 x 10^6 = 0

Make the substitution y = ^3 and divide out a 2:

8y^2 - (2000/)y - 96 x 10^6 = 0

Quadratic Formula:

y = (1/6) [2000/ +/- 1000(4/^2 + 1152)] = (1000/6) [2 +/- (4 + 1152^2)]

y = (1000/6) (2 + 106.648) = 5764 ==> = 35.8

r^2 = ^2 + 16 x 10^6 / ^4 = 321.475 + 16 x 10^6 / 103346 = 321.475 + 16 / (0.103346)

r^2 = 321.475 + 49.281 = 370.755 ==> r = 57.9 (Answer)

h = (r^2 - ^2) = (370.255 - 321.475) = 49.281 ==> h = 21 (Answer)

The silo should have a radius of 35.8, a height of 21.0, and the hemisphere should have a radius of 57.9

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