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#mathematics
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#rate-of-change
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#calculus
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#physics
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The diameter of a balloon is decreasing at the rate of 0.3cm/s while the height is increasing at a rate of 0.15cm/s. Find the rate at which the volume is changing when d=4cm and h =30cm?
Anthony Pitt
Mar 2
The diameter of a balloon is decreasing at the rate of 0.3cm/s while the height is increasing at a rate of 0.15cm/s. Find the rate at which the volume is changing when d=4cm and h =30cm?
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ChatGPT
Mar 2
To find the rate at which the volume is changing, we can use the formula for the volume of a sphere:
V = (4/3)πr^3
where r is the radius of the sphere. Since the diameter of the balloon is decreasing at a rate of 0.3 cm/s, the rate of change of the radius is half of that, which is -0.15 cm/s.
Given that d = 4 cm, the radius r = d/2 = 2 cm.
Now, we can differentiate the volume formula with respect to time t to find the rate at which the volume is changing:
dV/dt = 4πr^2(dr/dt)
Substitute the values we have:
dV/dt = 4π(2)^2(-0.15)
dV/dt = -3.6π
Therefore, the rate at which the volume is changing when d = 4 cm and h = 30 cm is -3.6π cm^3/s.
V = (4/3)πr^3
where r is the radius of the sphere. Since the diameter of the balloon is decreasing at a rate of 0.3 cm/s, the rate of change of the radius is half of that, which is -0.15 cm/s.
Given that d = 4 cm, the radius r = d/2 = 2 cm.
Now, we can differentiate the volume formula with respect to time t to find the rate at which the volume is changing:
dV/dt = 4πr^2(dr/dt)
Substitute the values we have:
dV/dt = 4π(2)^2(-0.15)
dV/dt = -3.6π
Therefore, the rate at which the volume is changing when d = 4 cm and h = 30 cm is -3.6π cm^3/s.
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