Advertisements
Advertisements
प्रश्न
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?
Advertisements
उत्तर
Let r cm be the radius, h cm be the height, S cm2 be the total surface area and V cm3 be the volume.
Now,
V = πr2h = 100
⇒ `h = 100/(pir^2)`
and S = 2πr2 + 2πrh
⇒ `S = 2pir^2 + 2pir (100/(pir^2))`
`= 2pir^2 + 200/r`
Differentiate `S = 2pir^2 + 200/r` w.r.t r we get
`(dS)/(dr) = 4pir - 200/r^2`
For maximum / minimum surface area
`(dS)/(dr) = 0`
⇒ `4pir - 200/r^2 = 0`
⇒ `r^3 = 200/ (4pi)`
⇒ `r = (50/pi)^(1//3)`
`(d^2S)/(dr^2) = 4pi + (200 xx 2)/r^3`
`= 4pi + 400/r^3`
and `((d^2S)/(dr^2))_(r = (50/pi)^(1/3))`
`= 4pi + 400/ (50/pi) > 0`
∴ S has a minimum value at
`r = (50/pi)^(1/3)`
When `r = (50/pi)^(1/3)` cm, then
`h = 100/(pi(50/pi)^(2//3)) `
`h = 100/((50)^(2//3) pi^(1//3))`
`= (50xx2)/ ((50)^(2//3) pi ^(1//3))`
`= 2 (50/pi)^(1//3)` cm.
When r `(50/pi)^(1/3)` and h = 2 `(50/pi)^(1/3)` then S will be minimum.
Hence, the total surface area will be minimum.
APPEARS IN
संबंधित प्रश्न
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
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 absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = sin x + cos x , x ∈ [0, π]
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum 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.
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 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.
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.
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 \[\pi : (\pi + 2)\].
Divide the number 30 into two parts such that their product is maximum.
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
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 : Show that of all rectangles inscribed in a given circle, the square has the maximum area.
Solve the following : A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.
A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?
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.
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The function `"f"("x") = "x" + 4/"x"` has ____________.
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 function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
Let f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______.
A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane bounded by the lines y = 0, y = 3x and y = 30 – 2x. The largest area of such a rectangle is ______.
The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
A running track of 440 m is to be laid out enclosing a football field. The football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum,then find the length of its sides. Also calculate the area of the football field.
Divide the number 100 into two parts so that the sum of their squares is minimum.
