Advertisements
Advertisements
प्रश्न
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)
Advertisements
उत्तर
i. Area of metal sheet required to made a cylinder open from top = 75π cm2.
Given, 'r' is the radius of the cylinder 'h' is the height of the cylinder.
∴ 2πrh + πr2 = 75π
`\implies` 2rh + r2 = 75
`\implies` 2rh = 75 – r2
`\implies` h = `(75 - r^2)/(2r)` ...(i)
Then, volume of cylinder,
ii. From (i),
V = πr2h
= `πr^2 xx ((75 - r^2)/(2r))` ...[From (i)]
= `(πr)/2 (75 - r^2) cm^3`.
V = `π/2 (75r - r^3)`
`(dV)/(dr) = π/2 (75 - 3r^2)` ...(ii)
iii. (a) From (ii),
`(dV)/(dr) = π/2 (75 - 3r^2)`
For maximum volume, put `(dV)/(dr)` = 0
`\implies π/2(75 - 3r^2)` = 0
`\implies (3pi)/2 != 0 , 25-r^2 = 0`
`\implies` 25 = r2 [after taking square roots]
`\implies` r = 5 cm
Now, `(d^2V)/(dr^2) = pi/2(– 3r^2) < 0`
Hence, the volume is maximum, when r = 5 cm.
OR
(b) [From ...(i)]
Then, h = `(75 - r^2)/(2r)`
`= (75-5^2)/(2(5))`
= `(75 - 25)/(10)`
= `50/10`
= 5
For max. volume,
h = r = 5 cm
Hence, the given statement is false.
APPEARS IN
संबंधित प्रश्न
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:
`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) = x3 − 6x2 + 9x + 15
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) = (x −1)2 + 3, x ∈[−3, 1]
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is `8/27` of the volume of the sphere.
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
Find the maximum area of an isosceles triangle inscribed in the ellipse `x^2/ a^2 + y^2/b^2 = 1` with its vertex at one end of the major axis.
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 \[\pi : (\pi + 2)\].
Find the maximum and minimum of the following functions : f(x) = x log x
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.
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
The function y = 1 + sin x is maximum, when x = ______
If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is `pi/3`
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
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 ______.
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 ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, 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.
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
