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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

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BuyFindarrow_forward

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Minimum Cost An industrial tank of the shape described in Exercise 33 must have a volume of 4000 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.

To determine

To calculate: The minimum cost of the tank.

Explanation

Given:

The tank is formed by attaching two hemispheres at the ends of a right circular cylinder. The total volume of the tank is 4000 cubic centimetres. Also, the cost for relaying the hemisphere is twice the cost of relaying the cylinder.

Formula used:

For a function f that is twice differentiable on an open interval I, if f'(c)=0 for some c, then,

If f''(c)>0 the function f has relative minima at c if f''(c)<0 the function f has relative maxima at c.

Calculation:

The hemisphere is attached the end of the cylinder and thus it would have the same radius as the cylinder.

Let the surface area of the solid be denoted by S and r denote the radius and h denote the height of the solid.

The surface area of the solid would sum of the lateral surface area of the cylinder and twice the lateral surface area of a hemisphere.

S=2πrh+2(2πr2)=2πrh+4πr2

Let the cost of relaying the cylinder be a units. This would give the total cost as:

C=a(2πrh)+2a(4πr2)=2aπrh+8aπr2

Now the volume of the provided tank would be the sum of the volume of the cylinder and the twice the volume of the hemisphere. The volume is provided as 4000 cubic centimetres.

Thus,

πr2h+2(2πr33)=4000πr2h+43πr3=4000πr2(h+4r3)=4000h=4000πr24r3

Substitute this in the cost function to obtain:

C(r)=2aπr(4000πr24r3)+8aπr2=a(8000r20π3r2)

Differentiate this with respect to r and equate it to 0

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