Advertisements
Advertisements
Question
The volume of a cylinder is given by the formula V = `pi"r"^2"h"`. Find the greatest and least values of V if r + h = 6
Advertisements
Solution
Given r + h = 6
⇒ r = 6 – h
Volume V = πr2h
V = π(6 – h)2h
`"dV"/"dh"` = π[(6 – h)2(1) + 2h(6 – h)(– 1)]
= π(6 – h)[6 – 3h]
For maximum or minimum,
`"dV"/"dh"` = 0
⇒ π(6 – h)(6 – 3h) = 0
⇒ h = 6, h = 2
h = 6 is not possible as r + h = 6
∴ h = 2
`("d"^2"V")/("dh"^2)` = π[(6 – h)(– 3) + (6 – 3h)(–1)]
= π[6h – 24]
At h = 2, `("d"^2"V")/("dh"^2) < 0`
∴ Volume of the cylinder is maximum when h = 2 and r = 6 – 2 = 4
Greatest value of V = π(4)2(2) = 32 π
Least value of V = 0
APPEARS IN
RELATED QUESTIONS
Find two positive numbers whose sum is 12 and their product is maximum
Find two positive numbers whose product is 20 and their sum is minimum
Find the smallest possible value of x2 + y2 given that x + y = 10
A garden is to be laid out in a rectangular area and protected by a wire fence. What is the largest possible area of the fenced garden with 40 meters of wire?
A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 1,80,000 sq. mtrs in order to provide enough grass for herds. No fencing is needed along the river. What is the length of the minimum needed fencing material?
Find the dimensions of the rectangle with maximum area that can be inscribed in a circle of radius 10 cm
Prove that among all the rectangles of the given perimeter, the square has the maximum area
Find the dimensions of the largest rectangle that can be inscribed in a semi-circle of radius r cm
A manufacturer wants to design an open box having a square base and a surface area of 108 sq.cm. Determine the dimensions of the box for the maximum volume
Find the asymptotes of the following curves:
f(x) = `x^2/(x + 1)`
Find the asymptotes of the following curves:
f(x) = `(x^2 - 6x - 1)/(x + 3)`
Choose the correct alternative:
One of the closest points on the curve x2 – y2 = 4 to the point (6, 0) is
