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A spherical balloon is filled with 3536π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 70π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 50 minutes after the leakage began is
- a)1/9
- b)1/2
- c)2/9
- d)35/18
Correct answer is option 'D'. Can you explain this answer?
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A spherical balloon is filled with 3536π cubic meters of helium gas. ...
Given:
- Volume of helium gas in the balloon = 3536π cubic meters
- Rate at which gas escapes = 70π cubic meters per minute
To find:
The rate at which the radius of the balloon decreases 50 minutes after the leakage began.
Assumption:
Assuming that the balloon is a perfect sphere, we can use the formula for the volume of a sphere to relate the volume and radius of the balloon.
Formula:
The volume of a sphere is given by V = (4/3)πr³, where V is the volume and r is the radius of the sphere.
Approach:
1. Find the initial radius of the balloon.
2. Find the initial rate at which the radius is changing.
3. Find the rate at which the radius changes after 50 minutes.
Calculation:
1. Initial radius of the balloon:
Using the formula V = (4/3)πr³, we can find the initial radius.
V = 3536π, so, 3536π = (4/3)πr³
Solving for r, we get r = cuberoot(3536*(3/4))
r = 14 meters
2. Initial rate of change of radius:
We know that the rate of change of volume with respect to time is given by dV/dt.
Since V = (4/3)πr³, we can differentiate this equation with respect to time to get dV/dt = 4πr²(dr/dt).
Given dV/dt = -70π cubic meters per minute (as the gas is escaping), and r = 14 meters (initial radius), we can solve for dr/dt.
-70π = 4π(14)²(dr/dt)
Solving for dr/dt, we get dr/dt = -70/(4*14²)
dr/dt = -35/98 meters per minute
3. Rate of change of radius after 50 minutes:
After 50 minutes, the rate at which the radius decreases will be the same as the initial rate of change of the radius: -35/98 meters per minute.
Therefore, the rate at which the radius of the balloon decreases 50 minutes after the leakage began is 35/98 meters per minute, which is equivalent to 35/18 meters per minute (after simplification).
Hence, the correct answer is option D) 35/18.
- Volume of helium gas in the balloon = 3536π cubic meters
- Rate at which gas escapes = 70π cubic meters per minute
To find:
The rate at which the radius of the balloon decreases 50 minutes after the leakage began.
Assumption:
Assuming that the balloon is a perfect sphere, we can use the formula for the volume of a sphere to relate the volume and radius of the balloon.
Formula:
The volume of a sphere is given by V = (4/3)πr³, where V is the volume and r is the radius of the sphere.
Approach:
1. Find the initial radius of the balloon.
2. Find the initial rate at which the radius is changing.
3. Find the rate at which the radius changes after 50 minutes.
Calculation:
1. Initial radius of the balloon:
Using the formula V = (4/3)πr³, we can find the initial radius.
V = 3536π, so, 3536π = (4/3)πr³
Solving for r, we get r = cuberoot(3536*(3/4))
r = 14 meters
2. Initial rate of change of radius:
We know that the rate of change of volume with respect to time is given by dV/dt.
Since V = (4/3)πr³, we can differentiate this equation with respect to time to get dV/dt = 4πr²(dr/dt).
Given dV/dt = -70π cubic meters per minute (as the gas is escaping), and r = 14 meters (initial radius), we can solve for dr/dt.
-70π = 4π(14)²(dr/dt)
Solving for dr/dt, we get dr/dt = -70/(4*14²)
dr/dt = -35/98 meters per minute
3. Rate of change of radius after 50 minutes:
After 50 minutes, the rate at which the radius decreases will be the same as the initial rate of change of the radius: -35/98 meters per minute.
Therefore, the rate at which the radius of the balloon decreases 50 minutes after the leakage began is 35/98 meters per minute, which is equivalent to 35/18 meters per minute (after simplification).
Hence, the correct answer is option D) 35/18.
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A spherical balloon is filled with 3536π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 70π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 50 minutes after the leakage began isa) 1/9b) 1/2c) 2/9d) 35/18Correct answer is option 'D'. Can you explain this answer?
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A spherical balloon is filled with 3536π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 70π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 50 minutes after the leakage began isa) 1/9b) 1/2c) 2/9d) 35/18Correct answer is option 'D'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about A spherical balloon is filled with 3536π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 70π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 50 minutes after the leakage began isa) 1/9b) 1/2c) 2/9d) 35/18Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A spherical balloon is filled with 3536π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 70π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 50 minutes after the leakage began isa) 1/9b) 1/2c) 2/9d) 35/18Correct answer is option 'D'. Can you explain this answer?.
A spherical balloon is filled with 3536π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 70π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 50 minutes after the leakage began isa) 1/9b) 1/2c) 2/9d) 35/18Correct answer is option 'D'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about A spherical balloon is filled with 3536π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 70π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 50 minutes after the leakage began isa) 1/9b) 1/2c) 2/9d) 35/18Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A spherical balloon is filled with 3536π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 70π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 50 minutes after the leakage began isa) 1/9b) 1/2c) 2/9d) 35/18Correct answer is option 'D'. Can you explain this answer?.
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