Advertisements
Advertisements
Question
Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi-vertical angle α is one-third that of the cone and the greatest volume of the cylinder is `(4)/(27) pi"h"^3 tan^2 α`.
Advertisements
Solution
Let CD = R, AD = x
⇒ OD = h - x
∵ ODC ∼ ΔOAB
⇒ `(h - x)/h = R/"AB" ⇒ (h - x)/h = R/(h tan α)`
⇒ R = (h - x) tan α
V = πR2x
= π(h - x)2 tan2α. x
= π tan2α (h - x)2x
`dV/dx = π tan2α (h2 - 4hx + 3x2)`
`dV/dx = 0 = h^2 - 4hx + 3x^2 = 0`
⇒ (h - x)(h - 3x) = 0
⇒ x = h (not possible ) or x =` h/3`
`(d^2V)/(dx^2) = π tan^2α (-4h + 6x)`
`((d^2V)/(dx^2))_(x = h/3) = π tan^2α (-2h) < 0`
⇒ V is maximum for x = `h/3`.
So, Vmax = π tan2α (h - x)2x
`= π tan^2α ( h - h/3 )^2 h/3`
= `(4πh)^3/27 tan^2 a`
APPEARS IN
RELATED QUESTIONS
Find the absolute maximum and absolute minimum values of the function f given by f(x)=sin2x-cosx,x ∈ (0,π)
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.
f (x) = 2x2 − 5x + 3 on [1, 3] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = x2 + 5x + 6 on the interval [−3, −2] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = cos 2x on [0, π] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x + cos x on [0, π/2] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = 4sin x on [0, π] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin4 x + cos4 x on \[\left[ 0, \frac{\pi}{2} \right]\] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x − sin 2x on [0, π]?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = x2 − 3x + 2 on [−1, 2] ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = 2x − x2 on [0, 1] ?
Discuss the applicability of Lagrange's mean value theorem for the function
f(x) = | x | on [−1, 1] ?
Using Lagrange's mean value theorem, prove that (b − a) sec2 a < tan b − tan a < (b − a) sec2 b
where 0 < a < b < \[\frac{\pi}{2}\] ?
State Rolle's theorem ?
If the value of c prescribed in Rolle's theorem for the function f (x) = 2x (x − 3)n on the interval \[[0, 2\sqrt{3}] \text { is } \frac{3}{4},\] write the value of n (a positive integer) ?
The value of c in Rolle's theorem for the function \[f\left( x \right) = \frac{x\left( x + 1 \right)}{e^x}\] defined on [−1, 0] is
Show that the local maximum value of `x + 1/x` is less than local minimum value.
Find the maximum and minimum values of f(x) = secx + log cos2x, 0 < x < 2π
Find the area of greatest rectangle that can be inscribed in an ellipse `x^2/"a"^2 + y^2/"b"^2` = 1
The maximum value of sinx + cosx is ______.
The least value of the function f(x) = `"a"x + "b"/x` (where a > 0, b > 0, x > 0) is ______.
The least value of the function f(x) = 2 cos x + x in the closed interval `[0, π/2]` is:
It is given that at x = 1, the function x4 - 62x2 + ax + 9 attains its maximum value on the interval [0, 2]. Find the value of a.
If f(x) = ax2 + 6x + 5 attains its maximum value at x = 1, then the value of a is
The function f(x) = [x], where [x] =greater integer of x, is
What is the greatest value of a function on the entire given interval called?
Which points are critical points for a function \[f\]?
