Answer to Question #174308 in Calculus for Holden Giles Cabrito

Question

A silo consists of a right circular cylinder surmounted by a right circular 

cone cover. What is the maximum volume of this silo that can be 

constructed using a 100 m2

of plane GI sheet ?

Answer 1

GIVEN:

Total surface area of given combination=area of given sheet

=S=lateral surface area of cone +curved surface area of cylinder +area of circle

S=$100=\pi rl+2\pi rh_2+\pi r^2\\$

as we know,

$l=\sqrt{r^2+h_1^2}$

and $l={r\over sin \theta}$

therefore,

$S=\pi r\sqrt{r^2+h_2^2}+2\pi rh_2+\pi r^2\\$ .......(1)

volume given by,

$V={1\over3}\pi r^2 h_1+\pi r^2h_2$ ........(2)

solving 1 and 2 we get,

$h_2=2h_1$

and then ,substituting this in (1),

for volume to be maximum, we diffrentiate V.

$\boxed{{dV\over dr}=0}$

we will get r=4 ,h₁₌ 3 ,h₂=6

using the following equation we will get maximum volume as,

350 m³ or say, (351.85)

This is the application of derivatives.