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
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
Find the maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
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 function. Find also the local maximum and the local minimum values, as the case may be:
`g(x) = 1/(x^2 + 2)`
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) = 4x - 1/x x^2, x in [-2 ,9/2]`
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening
A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.
Show that the minimum length of the hypotenuse is `(a^(2/3) + b^(2/3))^(3/2).`
Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has
- local maxima
- local minima
- point of inflexion
Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π].
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
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].
The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.
Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x
Find the maximum and minimum of the following functions : f(x) = `logx/x`
Solve the following : A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.
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.
Divide the number 20 into two parts such that their product is maximum
By completing the following activity, examine the function f(x) = x3 – 9x2 + 24x for maxima and minima
Solution: f(x) = x3 – 9x2 + 24x
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme values, f'(x) = 0, we get
x = `square` or `square`
∴ f''`(square)` = – 6 < 0
∴ f(x) is maximum at x = 2.
∴ Maximum value = `square`
∴ f''`(square)` = 6 > 0
∴ f(x) is maximum at x = 4.
∴ Minimum value = `square`
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
An open box with square base is to be made of a given quantity of cardboard of area c2. Show that the maximum volume of the box is `"c"^3/(6sqrt(3))` cubic units
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/cm2 and the material for the sides costs Rs 2.50/cm2. Find the least cost of the box.
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
The maximum value of `(1/x)^x` is ______.
A ball is thrown upward at a speed of 28 meter per second. What is the speed of ball one second before reaching maximum height? (Given that g= 10 meter per second2)
The function `f(x) = x^3 - 6x^2 + 9x + 25` has
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
The maximum value of the function f(x) = `logx/x` is ______.
Let P(h, k) be a point on the curve y = x2 + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is ______.
A cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is ______.
Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
A running track of 440 m is to be laid out enclosing a football field. The football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum,then find the length of its sides. Also calculate the area of the football field.
A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.
Mrs. Roy designs a window in her son’s study room so that the room gets maximum sunlight. She designs the window in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the window that will admit maximum sunlight into the room.
A maximum or minimum value of \[f\] is called a(n) _____.
When is a point in the domain of a function called a critical point?
Which statement gives the meaning of a local maximum at \[c\]?
Why must the left endpoint \[a\] and right endpoint \[b\] be checked for absolute extrema on \[[a,b]\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], which is the correct factorization of \[f'(x)\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=-2\]?
