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.
dV = surface area * dr
so
dr/dt = dV/dt/(4 pi r^2)
dr/dt = 2 /(4 pi *64)
To find how fast the radius of the balloon is shrinking, we can use the formula for the volume of a sphere:
V = (4/3) * π * r^3
where V is the volume and r is the radius.
We know that the balloon is losing air at a rate of 2 cubic inches per minute, so the rate of change of the volume with respect to time is:
dV/dt = -2
To find how fast the radius is changing, we need to differentiate the volume equation with respect to time:
dV/dt = (dV/dr) * (dr/dt)
to solve for (dr/dt), the rate of change of the radius with respect to time.
Let's differentiate the volume equation:
dV/dt = 4πr^2 * (dr/dt)
Now substitute the given rate of change of volume and the radius:
-2 = 4π(8^2) * (dr/dt)
Simplifying:
-2 = 256π * (dr/dt)
Now solve for (dr/dt), the rate at which the radius is shrinking:
(dr/dt) = -2 / (256π)
≈ -0.00124 inches per minute
Therefore, the radius of the balloon is shrinking at a rate of approximately 0.00124 inches per minute when the radius is 8 inches.
To find how fast the radius of the balloon is shrinking, we can apply the chain rule from calculus. The volume of a sphere is given by the formula V = (4/3) * π * r^3, where V represents the volume and r is the radius.
Differentiating this formula with respect to time (t) gives us:
dV/dt = (4/3) * π * (3r^2) * (dr/dt)
Here, dV/dt represents the rate at which the volume is changing with respect to time, and dr/dt represents the rate at which the radius is changing with respect to time.
We are given that the volume is changing at a rate of -2 cubic inches per minute, so dV/dt = -2. We also know that the initial radius is 8 inches, so r = 8.
Plugging these values into the equation, we can solve for dr/dt:
-2 = (4/3) * π * (3 * 8^2) * (dr/dt)
Simplifying further:
-2 = (4/3) * π * 192 * (dr/dt)
To find dr/dt, we rearrange the equation and solve for the rate of change of the radius:
dr/dt = -2 / ((4/3) * π * 192)
Evaluating this expression gives us the specific rate at which the radius is shrinking when the radius is 8 inches.