KVPY SX SB Exam 05-Nov-2017 Question Paper

Question : 83 of 120

Marks: +1, -0

A solid hemisphere is mounted on a solid cylinder, both having equal radii. If the whole solid is to have a fixed surface area and the maximum possible volume, then the ratio of the height of the cylinder to the common radius is

1:1
1:2
2:1
$√ 2 :1$

Validate

Solution:

S = 2πR^ 2 + 2πRh + πR^ 2

(R = radius of hemisphere & cylinder)

V=2/3πR^ 3 + πR^ 2h

V=2/3πR^ 3 + πR^ 2×({5-3πR^2}/{2πR})

{dV}/{dR}=2πR^2+5/2-{9π}/2R^2

for maximum & minimum

{dV}/{dR}=0

5πR^22 = S

5πR^2 = 3πR^2 + 2πRh

2R = 2h
h : R = 1 : 1

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