Advertisements
Advertisements
प्रश्न
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
Advertisements
उत्तर
Let x be the side of square base of cuboid and other side be y.
Then the volume of a cuboid with square base,
V = x × x × y
⇒ V = x2y
As the volume of the cuboid is given so volume is taken constantly throughout the question, therefore,
y = `"V"/"x"^2` ....(i)
In order to show that surface area is minimum when the given cuboid is a cube, we have to show S” > 0 and x = y.
Let S be the surface area of cuboid, then
S = x2 + xy + xy + xy + xy + x2
S = 2x2 + 4xy .....(ii)
⇒ S = 2x2 + 4x. `"V"/"x"^2`
⇒ S = 2x2 + `"4V"/"x"` ....(iii)
⇒ `"dS"/"dx" = "4x" - "4V"/"x"^2` ....(iv)
For maximum/minimum value of S, we have `"dS"/"dx" = 0`
⇒ `4"x" - "4V"/"x"^2 = 0 => 4"V" = 4"x"^3`
⇒ V = x3 ....(v)
Putting V = x3 in (i) , we have
y = `"x"^3/"x"^2 = "x"`
Here, y = x ⇒ cuboid is a cube.
Differentiating (iv) w.r.t.x, we have
`("d"^2"S")/"dx"^2 = (4 +(8"V")/"x"^3) >0`
Hence, surface area is minimum when given cuboid is a cube.
APPEARS IN
संबंधित प्रश्न
Find the maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 2
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
Choose the correct option from the given alternatives :
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.
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.
Divide the number 20 into two parts such that their product is maximum
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
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 ______.
The maximum value of `(1/x)^x` is ______.
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 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 ____________.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
Divide 20 into two ports, so that their product is maximum.
The range of a ∈ R for which the function f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2), x ≠ 2nπ, n∈N` has critical points, is ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.
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)
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
If Mr. Rane order x chairs at the price p = (2x2 - 12x - 192) per chair. How many chairs should he order so that the cost of deal is minimum?
Solution: Let Mr. Rane order x chairs.
Then the total price of x chairs = p·x = (2x2 - 12x- 192)x
= 2x3 - 12x2 - 192x
Let f(x) = 2x3 - 12x2 - 192x
∴ f'(x) = `square` and f''(x) = `square`
f'(x ) = 0 gives x = `square` and f''(8) = `square` > 0
∴ f is minimum when x = 8
Hence, Mr. Rane should order 8 chairs for minimum cost of deal.
If \[\mathrm{A}+\mathrm{B}=\frac{\pi}{2}\] then the maximum value of cosA.cosB is
The absolute maximum value of the function f(x) = 2x3 − 3x2 − 36x + 9 defined on [−3, 3] is ______.
