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 approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.
If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.
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`
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.`
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
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.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
Solve the following : Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.
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)`.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
The function f(x) = x log x is minimum at x = ______.
A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum
Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is ______
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.
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 smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
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.
Read the following passage and answer the questions given below.
|
In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of `x^2/a^2 + y^2/b^2` = 1. |
- If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
- Find the critical point of the function.
- Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
OR
Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
A function f(x) is maximum at x = a when f'(a) > 0.
Let f: R → R be a function defined by f(x) = (x – 3)n1(x – 5)n2, n1, n2 ∈ N. Then, which of the following is NOT true?
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 ______.
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.
Assume \[f'(c)=0\] and the second derivative exists at \[c\]. Which condition gives a local maximum?
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 is \[f''(x)\]?
