Home
Class 12
MATHS
A spherical balloon is filled with 4500p...

A spherical balloon is filled with 4500p cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of `72pi` cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is (1) `9/7` (2) `7/9` (3) `2/9` (4) `9/2`

A

`6//7`

B

`4//9`

C

`2//9`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
C
`V=4/3pir^(3)`
Initially volume of gas in balloon is 4500 `pi` cubic meters.
`therefore 4500 pi=(4pir^(3))/(3)`
`therefore r=15` m.
Given `(dV)/(dt) = 72pi` cubic meter per minute
`therefore` After 49 minutes, volume of gas leaked
`=72pi xx 49`
`therefore` Volume of gas in balloon after 49 minutes
`=(4500-49xx72)pi=972pim^(3)`
`therefore 972pi=(4pir^(3))/(3)`
`therefore r^(3)=3 xx 243 = 3 xx 3^(5)`
`therefore r=9`
Now, `(dV)/(dt) = 4pir^(2) (dr)/(dt)`
For the rate of decrease in radius after 49 minutes.
`72pi = 4pi xx 9 xx 9(dr)/(dt)`
`(dr)/(dt) 2/9` meter per minute
Promotional Banner

Topper's Solved these Questions

  • DIFFERENTIAL EQUATIONS

    CENGAGE|Exercise JEE Advanced Previous Year|12 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE|Exercise Single Correct Answer Type|37 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE|Exercise Exercise (Numerical)|15 Videos

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE|Exercise Exercise|337 Videos

  • DIFFERENTIATION

    CENGAGE|Exercise Numerical Value Type|3 Videos

Similar Questions

Explore conceptually related problems

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.

If we blow air into a balloon of spherical shape at a rate of 1000 cm^(3) per second. At what rate the radius of the balloon changes when the radius is 7cm? Also compute the rate at which the surface area changes .

At 27^(@) C , H_(2) is leaked through a tiny hole into a vessel for 20 minutes Another unknown gas at the same T and P as that of H_2 is leaked through the same hole for 20 minutes. After effusion of the gas, the mixture exerts a pressure of 6 atm. The H_2 content of the mixture is 0.7 moles. If volume of the container is 3 litres what is the molecular weight of unknown gas ?

The decomposition of Cl_(2)O_(7) at 500 K in the gas phase to Cl_(2) and O_(2) is a first order reaction. After 1 minute at 500 K, the pressure of Cl_(2)O_(7) falls from 0.08 to 0.04 atm. Calculate the rate constant in s^(-1) .

If pa n dq are chosen randomly from the set {1,2,3,4,5,6,7,8,9, 10} with replacement, determine the probability that the roots of the equation x^2+p x+q=0 are real.

Displacement of a particle is given by the expression x = 3 t ^ { 2 } + 7 t - 9 , where x is in meter and t is in seconds. What is acceleration ?

Two sides of a rhombus are along the lines, x-y+1=0 and 7x-y-5=0 . If its diagonals intersect at (-1,-2) , then which one of the following is a vertex of this rhombus ? (1) (-3,-9) (2) (-3,-8) (3) (1/3,-8/3) (4) (-(10)/3,-7/3)

Let P=0 be the equation of a plane passing through the line of intersection of the planes 2x-y=0a n d3z-y=0 and perpendicular to the plane 4x+5y-3z=8. Then the points which lie on the plane P=0 is/are a. (0,9,17) b. (1//7,21//9) c. (1,3,-4) d. (1//2,1,1//3)