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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

A metal storage tank with volume V is to be constructed in the shape of a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal?

To determine

To find:

The dimensions that gives the least amount of metal

Explanation

1) Concept:

i) The function has local minima when f'x<0, and has local maxima when f'x>0

2) Formula:

i. Power rule for xn is given by 

ddxxn=nxn-1

ii. Volume of the cylinder =πr2h

iii. Volume of the hemisphere =23πr3

iv. Surface area of cylinder (lateral) = 2πrh

v. Surface area of hemisphere = 124πr2

vi. Surface area of circle = πr2

3) Calculation: From the given information, the required cylinder looks like

The storage tank has two parts - the hemisphere and the cylinder.

Let h be the length and r be the radius of the cylinder.

Let the volume of the tank be V.

Minimize the surface area so that least amount of metal is used

The surface area of the tank is sum of the surface area of the hemisphere, the cylinder, and the area of the bottom circle. Therefore, S=124πr2+2πrh+πr2

Combine like terms.

S=3πr2+2πrh

Now volume of the tank is sum of the volume of the cylinder and volume of the hemisphere.

V=23πr3+πr2h

From this find the value of h

V-23πr3=πr2h

h=Vπr2-23πr3πr2=Vπr2-23r

Substitute this value of h in the surface area S=3πr2+2πrh

S=3πr2+2πrVπr2-23r

S=2Vr+53πr2

This is the function of r so,

S(r)=2Vr+53πr2

To find the minimum value of S differentiate S(r)

S'(r)=-2Vr2+103πr

Equate this with zero and find the critical point

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