You are letting the air out of a hot air balloon at a steady rate. The volume of the balloon is decreasing at the rate of 10pi m^3/min when the diameter is 45 meters. At what rate is the radius of the balloon changing? I need help! How do I solve this problem? Thank you so much for the help!
V = (4/3)pi(r^3)
so dV/dt = 4pi(r^2)dr/dt
so when diameter is 45 m
r = 22.5, dV/dt = -10pi
plug that in to solve for dr/dt
Thanks, that's helpful. Would the answer be .20m/min?
I had
-10pi = 4pi(22.5)^2 dr/dt
dr/dt = -10/(4(22.5)^2)
= - 0.00494 m/min
or -.494 cm/min
seems like a small amount, but remember that balloon is huge, so a small change in the radius is going to result in a large change in the volume.
BTW, the negative indicates that the radius is decreasing.
Thanks, I forgot to divide the 4. The negative makes sense.
To solve this problem, we can use the relationship between the volume of a sphere and its radius. The volume of a sphere is given by the formula V = (4/3)πr^3, where V represents the volume and r represents the radius.
We know that the volume of the balloon is decreasing at a rate of 10π m^3/min. Let's denote this rate as dV/dt. Thus, dV/dt = -10π m^3/min since the volume is decreasing.
We are asked to find the rate at which the radius of the balloon is changing, which is represented by dr/dt. We need to find a relationship between dV/dt and dr/dt.
First, differentiate the volume equation with respect to time (t) using the chain rule:
dV/dt = (dV/dr)*(dr/dt)
The derivative dV/dr gives us the rate of change of volume with respect to the radius, which is the derivative of (4/3)πr^3 with respect to r. So, dV/dr = 4πr^2.
Substituting these values into the equation, we now have:
-10π m^3/min = (4πr^2)*(dr/dt)
To find dr/dt, we can rearrange the equation:
dr/dt = (-10π)/(4πr^2)
Now, we can plug in the given information, which is the diameter of the balloon (45 meters). The radius (r) is half the diameter, so r = 45/2 = 22.5 meters.
Substituting the value of r into the equation, we have:
dr/dt = (-10π)/(4π(22.5)^2)
Simplifying further:
dr/dt = (-10π)/(4π(506.25))
dr/dt = (-10π)/(2025π)
dr/dt = -10/2025
Therefore, the rate at which the radius of the balloon is changing is -10/2025 meters per minute.