is my answer to this question correct
A spherical balloon is losing air at the rate of 2 cubic inches per minute. How fast is the radius of the ballon shrinking when the radius is 8 inches.
Answer= .0024
V=4/3 PI r^3
dV/dt=4/3 PI * 3r^2 dr/dt
solve for dr/dt
dr/dt=dV/dt / 4PIr^2
Wondering why you did not round it up to .0025
To determine if your answer is correct, we need to use the information given and apply the appropriate formulas.
We know that the balloon is losing air at a rate of 2 cubic inches per minute. We also need to find out how the radius is changing.
The volume of a sphere is given by the formula V = (4/3) * π * r^3, where V is the volume and r is the radius.
To find how fast the radius is shrinking, we need to find dV/dt, the rate at which the volume of the balloon is changing with respect to time. We can then use this to find dr/dt, the rate at which the radius is changing with respect to time.
Differentiating the volume formula with respect to time, we get:
dV/dt = d/dt[(4/3) * π * r^3]
= (4/3) * π * d/dt(r^3)
= (4/3) * π * 3r^2 * dr/dt
= 4πr^2 * dr/dt
Now, we know that dV/dt = -2, since the balloon is losing air at a rate of 2 cubic inches per minute. We also know that r = 8 inches.
Substituting these values into the equation, we get:
-2 = 4π(8^2) * dr/dt
Simplifying further:
-2 = 4π(64) * dr/dt
-2 = 256π * dr/dt
Finally, we can solve for dr/dt:
dr/dt = -2 / (256π)
= -1 / (128π)
≈ -0.002448
Therefore, the correct answer is approximately -0.002448. It seems that your answer of 0.0024 is incorrect.