Chapter 3.R, Problem 46E
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Calculus (MindTap Course List)
8th Edition
James Stewart
ISBN: 9781285740621
Solutions
Chapter
Section
BuyFindarrow_forward
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
2) Formula:
i. Power rule for
ii. Volume of the cylinder
iii. Volume of the hemisphere
iv. Surface area of cylinder (lateral)
v. Surface area of hemisphere
vi. Surface area of circle
3) Calculation: From the given information, the required cylinder looks like
The storage tank has two parts - the hemisphere and the cylinder.
Let
Let the volume of the tank be
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,
Combine like terms.
Now volume of the tank is sum of the volume of the cylinder and volume of the hemisphere.
From this find the value of
Substitute this value of
This is the function of
To find the minimum value of
Equate this with zero and find the critical point
Chapter 3 Solutions
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