Advertisements
Advertisements
प्रश्न
Solve the following : Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.
Advertisements
उत्तर
Given the right circular cone of fixed height h and semi-vertical angle `oo`.
Let R be the radius of the base and H be the height of right circular cylinder that can be inscribed in the right circular cone.
In the figure, ∠GAO = ∞, OG = r, OA = h, OE = R, CE = H.
We have `r/h = tanx`
∴ r = h tan x ...(1)
Since ΔAOG anf ΔCEG are similar.
∴ `"AO"/"OG" = "CE"/"EG" = "CE"/"OG - OE"`
∴ `h/r = "H"/"r - R"`
∴ H = `h/r(r - R)`
= `"h"/"h tan oo"(h tan oo- R)` ...[By (1)]
∴ H = `(1)/tanoo(h tan oo - R)` ...(2)
Let V be the volume of the cylinder
Then V = `piR^2H = (piR^2)/tanoo(h tan oo - R)`
∴ V = `piR2h - (piR^3)/tan oo`
∴ `"dV"/"dR" = d/"dR"(piR^2h - (piR^3)/tan oo)`
= `piR xx 2R - pi/tan oo xx 3R^2`
= `2piRh - (3piR^2)/tan oo`
and
`(d^2V)/(dR^2) = d/"dR"(2piRh - (3pir^2)/tan oo)`
= `2pih xx 1 - (3pi)/tanoo xx 2R`
= `2pih - (6piR)/tanoo`
For maximum volume, `"dV"/"dR"` = 0
∴ `2piRh - (3piR^2)/tanoo` = 0
∴ `(3piR2)/tanoo = 2piRh`
∴ R = `(2h)/(3) tanoo` ...[∵ R ≠ 0]
and
`((d^2V)/(dR^2))_("at" R = (2h)/(3)tanx`
= `2pih - (6pi)/tanoo xx (2h)/(3) tan oo`
= 2πh – 4πh = – 2πh < 0
∴ V is maximum when R = `(2h)/(3)tanoo`
When R = `(2h)/(3)tanoo`, then from (2), we get
H = `(1)/tanoo(h tan oo - (2h)/3 tann oo) = h/(3)`
Hence, the height of the right circular cylinder is one- third of that of the cone.
APPEARS IN
संबंधित प्रश्न
An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere
A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.
Find the maximum and minimum value, if any, of the following function given by g(x) = − |x + 1| + 3.
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
`h(x) = sinx + cosx, 0 < x < pi/2`
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) = sinx − cos x, 0 < x < 2π
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) = x3 − 6x2 + 9x + 15
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
`f(x) = xsqrt(1-x), x > 0`
Prove that the following function do not have maxima or minima:
g(x) = logx
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) =x^3, x in [-2,2]`
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = sin x + cos x , x ∈ [0, π]
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3.`
A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5 per cm2 and the material for the sides costs Rs 2.50 per cm2. Find the least cost of the box
Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.
Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
Divide the number 20 into two parts such that sum of their squares is minimum.
A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.
Solve the following:
A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.
Solve the following : Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is `(2"R")/sqrt(3)`. Also, find the maximum volume.
Determine the maximum and minimum value of the following function.
f(x) = 2x3 – 21x2 + 36x – 20
The function f(x) = x log x is minimum at x = ______.
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
If z = ax + by; a, b > 0 subject to x ≤ 2, y ≤ 2, x + y ≥ 3, x ≥ 0, y ≥ 0 has minimum value at (2, 1) only, then ______.
Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is ______
The function y = 1 + sin x is maximum, when x = ______
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.
The maximum value of sin x . cos x is ______.
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
Find the points of local maxima and local minima respectively for the function f(x) = sin 2x - x, where `-pi/2 le "x" le pi/2`
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
Find both the maximum and minimum values respectively of 3x4 - 8x3 + 12x2 - 48x + 1 on the interval [1, 4].
The combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R1 + R2 = C (a constant), then maximum resistance R is obtained if ____________.
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane bounded by the lines y = 0, y = 3x and y = 30 – 2x. The largest area of such a rectangle is ______.
The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° is ______.
A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
Read the following passage:
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore.
|
Based on the above information, answer the following questions:
- If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
- Find `(dV)/(dr)`. (1)
- (a) Find the radius of cylinder when its volume is maximum. (2)
OR
(b) For maximum volume, h > r. State true or false and justify. (2)
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
