Advertisements
Advertisements
प्रश्न
Solve the following : 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)`.
Advertisements
उत्तर
Let x be the radius of base and h be the height of the cone which is inscribed in a sphere of radius r.
In the figure, AD = h and CD = x = BD
Since, ΔABD and ΔBDE are similar,
`"AD"/"BD" = "BD"/"DE"`
∴ BD2 = AD.DE = AD.(AE – AD)
∴ x2 = h(2r – h) ...(1)
Let V be the volume of the cone.
Then V = `(1)/(3)pix^2h`
= `pi/(3)h(2r - h)h` ...[By (1)]
∴ V = `pi/(3)(2rh^2 - h^3)`
∴ `"dV"/"dh" = pi/(3) d/"dh"(2rh^2 - h^3)`
= `pi/(3)(2r xx 2h - 3h^2)`
= `pi/(3)(4rh - 3h^2)`
and
`(d^2V)/(dh^2) = pi/(3).d/"dh"(4rh - 3h^2)`
= `pi/(3)(4r xx 1 - 3 xx 2h)`
= `pi/(3)(4r - 6h)`
For maximm volume, `"dV"/"dh"` = 0
∴ `pi/(3)(4rh - 3h^2)` = 0
∴ `4rh = 3h^2`
∴ h = `(4r)/(3)` ...[∵ h ≠ 0]
and
`((d^2V)/"dh"^2)_("at" h = (4r)/(3)`
= `pi/(3)(4r - 6 xx (4r)/3)`
= `pi/(3)(4r - 8r)`
= `-(4pir)/(3) < 0`
∴ V is maximum when h = `(4r)/(3)`
Hence, the attitude (i.e. height) of the right circular cone of maximum volume = `(4r)/(3)`.
APPEARS IN
संबंधित प्रश्न
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
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) = x/2 + 2/x, x > 0`
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
Show that the right circular cone of least curved surface and given volume has an altitude equal to `sqrt2` time the radius of the base.
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
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
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .
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 \[\pi : (\pi + 2)\].
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
Find the maximum and minimum of the following functions : f(x) = `logx/x`
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
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.
Solve the following:
A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
Divide the number 20 into two parts such that their product is maximum
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
The function y = 1 + sin x is maximum, when x = ______
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
Find the points of local maxima, local minima and the points of inflection of the function f(x) = x5 – 5x4 + 5x3 – 1. Also find the corresponding local maximum and local minimum values.
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
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume.
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
If y `= "ax - b"/(("x" - 1)("x" - 4))` has a turning point P(2, -1), then find the value of a and b respectively.
If y = x3 + x2 + x + 1, then y ____________.
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 ____________.
Read the following passage and answer the questions given below.
|
|
- Is the function differentiable in the interval (0, 12)? Justify your answer.
- If 6 is the critical point of the function, then find the value of the constant m.
- Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.
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 minimum value of 2sinx + 2cosx is ______.
A straight line is drawn through the point P(3, 4) meeting the positive direction of coordinate axes at the points A and B. If O is the origin, then minimum area of ΔOAB is equal to ______.
Find the maximum and the minimum values of the function f(x) = x2ex.
If \[\mathrm{A}+\mathrm{B}=\frac{\pi}{2}\] then the maximum value of cosA.cosB is
If \[f\] has a maximum value at \[c\], what is \[c\] called?
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?
Assume \[f'(c)=0\] and the second derivative exists at \[c\]. Which condition gives a local minimum?
Why must the left endpoint \[a\] and right endpoint \[b\] be checked for absolute extrema on \[[a,b]\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what is \[f''(x)\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=0\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=-2\]?
