Cost of Industrial Tank?
An industrial tank shaped by two hemispheres adjoining to the ends of a right circular cyliner has a colume of 3000 cubic feet. The hemispherical ends cost twice as much per square foot of surface area as the sides. Find the dimensions that will minimize cost! help!
The question is a little ambiguous… Do the two ends TOGETHER cost twice as much as the sides, or does EACH end cost twice as much as the sides?
Anyway, the formula for volume for this figure is:
V = (4/3)(pi)(r^3) + (pi)(r^2)(x)
where r is the radius of the spherical ends and x is the length of the cylinder.
The surface area of one of the hemispheres is 2(pi)(r^2) and the surface area of the cylinder part is (2r)(pi)(x).
So…I’m going to assume that EACH end is twice the cost of the cylinder…
(2)(pi)(r^2) = 2(2r)(pi)(x)
Simplify this and solve for r:
r^2 = 2rx
r^2 – 2rx = 0
r(r-2x) = 0
so, r=0 (impossible) or r=2x
——-
Now, put 2x in the volume equation:
V = (4/3)(pi)(r^3) + (pi)(r^2)(x)
3000 = (4/3)(pi)(2x)^3 + (pi)(x)(2x)^2
I’m going to multiply everything by 3 to get rid of the fraction:
9000 = (4)(pi)(8x^3) + (3)(pi)(x)(4x^2)
9000 = 32(pi)(x^3) + 12(pi)(x^3)
2250 = 8(pi)(x^3) + 3(pi)(x^3)
2250 = 11(pi)(x^3)
65.1088 = x^3
4.02297 = x
So the length of the cylinder is 4 ft.
You can use that to find the radius of the hemisphere:
r = 2x = 2(4) = 8
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