a spherical balloon is inflated with gas at the rate of 500 cubic centimeters per minute. how fast is the radius of the balloon increasing at the instant the radius is 30 centimeters?
The vloume increase rate is dV/dt = 500
V = (4/3) pi R^3
dV/dt = (12/3)pi R^2 dR/dt
= 4 pi R^2 dR/dt
You can use that equation to compute the dR/dt expansion rate for any value of R.
Note that that dV/dt equals the instantaneous surface area times dR/dt.
r=30cm
500=4 pi R^2 dR/dt
500=11309.7336 dR/dt
dR/dt = .0442cm/min
To find the rate at which the radius of the balloon is increasing at a specific instant, we can use the equation for the volume of a sphere:
V = (4/3)πr^3,
where V is the volume of the balloon and r is the radius.
The problem states that the balloon is being inflated at a rate of 500 cubic centimeters per minute, which means:
dV/dt = 500 cubic centimeters per minute.
To find how the radius of the balloon is increasing, we need to find dr/dt, the rate at which the radius is changing. To do this, we can differentiate the equation for the volume of a sphere with respect to time:
dV/dt = (4/3)π(3r^2)(dr/dt).
Substituting the given value of dV/dt and the radius r = 30 cm, we have:
500 = (4/3)π(3(30)^2)(dr/dt).
Now we can solve for dr/dt:
dr/dt = (500) / (4π(30)^2/3).
Simplifying this expression gives you the rate at which the radius is increasing at the instant the radius is 30 centimeters.
To find how fast the radius of the balloon is increasing, we can use the formula for the volume of a sphere. The volume (V) of a sphere is given by V = (4/3)πr^3, where r is the radius of the sphere.
If the balloon is inflating at a rate of 500 cubic centimeters per minute, we can determine the rate of change of the volume with respect to time. Let's call this rate dV/dt.
We can differentiate the volume formula with respect to time using the chain rule.
dV/dt = d((4/3)πr^3)/dt
To differentiate this equation, we need to differentiate each term with respect to time.
dV/dt = (4/3)π * d(r^3)/dt
Let's determine d(r^3)/dt. We know that r is the radius of the balloon, which is changing with time. So we differentiate r^3 with respect to time.
d(r^3)/dt = 3r^2 * dr/dt
Now, we substitute this back into the original equation:
dV/dt = (4/3)π * 3r^2 * dr/dt
Simplifying further:
dV/dt = 4πr^2 * dr/dt
Now, we have an equation that relates the rate of change of the volume with respect to time (dV/dt) to the rate of change of the radius with respect to time (dr/dt). We can solve for dr/dt by rearranging this equation:
dr/dt = (dV/dt) / (4πr^2)
Now, we know that dV/dt is given as 500 cubic centimeters per minute and the radius (r) is given as 30 centimeters at the instant we are interested in.
dr/dt = (500 cm^3/min) / (4π(30 cm)^2)
Using this equation, you can plug in the values and calculate the rate at which the radius of the balloon is increasing at the instant when the radius is 30 centimeters.