Advertisements
Advertisements
प्रश्न
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
Advertisements
उत्तर
Let R be the radius and h be the height of the cylinder which is inscribed in a sphere of radius r cm.
Then from the figure,
`"R"^2 + (h/2)^2` = r2
∴ R2 = `r^2 - h^2/(4)` ...(1)
Let V be the volume of the cylinder.
Then V = πR2h
= `pi(r^2 - h^2/(4))h` ...[By (1)]
= `pi(r^2 - h^3/(4))`
∴ `"dV"/"dh" = pid/"dh"(r^2h - h^3/(4))`
= `pi(r^2 xx 1 - 1/4 xx 3h^2)`
= `pi(r^2 - 3/4h^2)`
and
`(d^2V)/("dh"^2) = pid/"dh"(r^2 - 3/4h^2)`
= `pi(0 - 3/4 xx 2h)`
= `-(3)/(2)pih`
Now, `"dV"/"dh" = 0 "gives", pi(r^2 - 3/4h^2)` = 0
∴ `r^2 - 3/4h^2` = 0
∴ `(3)/(4)h^2` = r2
∴ h2 = `(4r^2)/(3)`
∴ h = `(2r)/sqrt(3)` ...[∵ h > 0]
and
`((d^2V)/(dh^2))_("at" h = (2r)/sqrt(3)`
= `-(3)/(2)pi xx (2r)/sqrt(3) < 0`
∴ V is maximum at h = `(2r)/sqrt(3)`
If h = `(2r)/sqrt(3)`, then from (1)
R2 = `r^2 - (1)/(4) xx (4r^2)/(3) = (2r^2)/(3)`
∴ volumeof the largest cylinder
= `pi xx (2r^2)/(3) xx (2r)/sqrt(3) = (4pir^3)/(3sqrt(3)`cu cm.
Hence, the volume of the largest cylinder inscribed in a sphere of radius 'r' cm = `(4pir^3)/(3sqrt(3)`cu cm.
APPEARS IN
संबंधित प्रश्न
Find the maximum and minimum value, if any, of the following function given by f(x) = |sin 4x + 3|
Find the maximum and minimum value, if any, of the following function given by h(x) = x + 1, x ∈ (−1, 1)
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) = x2
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) = x3 − 3x
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)`
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 both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
What is the maximum value of the function sin x + cos x?
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.
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(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
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
Find the point on the straight line 2x+3y = 6, which is closest to the origin.
Find the maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x.
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.
Show that among rectangles of given area, the square has least perimeter.
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a2. Show that the maximum volume of the box is `a^3/(6sqrt(3)`.
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
The function f(x) = x log x is minimum at x = ______.
A metal wire of 36 cm long is bent to form a rectangle. By completing the following activity, find it’s dimensions when it’s area is maximum.
Solution: Let the dimensions of the rectangle be x cm and y cm.
∴ 2x + 2y = 36
Let f(x) be the area of rectangle in terms of x, then
f(x) = `square`
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme value, f'(x) = 0, we get
x = `square`
∴ f''`(square)` = – 2 < 0
∴ Area is maximum when x = `square`, y = `square`
∴ Dimensions of rectangle are `square`
The minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
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 ______.
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
AB is a diameter of a circle and C is any point on the circle. Show that the area of ∆ABC is maximum, when it is isosceles.
The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `x/3` and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.
If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.
If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
If the point (1, 3) serves as the point of inflection of the curve y = ax3 + bx2 then the value of 'a ' and 'b' are ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
The set of values of p for which the points of extremum of the function f(x) = x3 – 3px2 + 3(p2 – 1)x + 1 lie in the interval (–2, 4), is ______.
The lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, is ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` 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.
Check whether the function f : R `rightarrow` R defined by f(x) = x3 + x, has any critical point/s or not ? If yes, then find the point/s.
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
A maximum or minimum value of \[f\] is called a(n) _____.
What is an absolute maximum?
Let \[c\] be a critical point of a continuous function \[f\]. If \[f'(x)\] changes sign from positive to negative as \[x\] passes through \[c\], what is \[c\]?
What are the critical points obtained from \[12x(x-1)(x+2)=0\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=1\]?
