If you are blowing up a balloon at a rate of 3 cubic ft per min, what is the rate of change of the radius after 30 seconds
dV/dt = 3ft^3/min(1 min/60s) = .05 ft^3/s
so
V = .05 * 30 = 1.5 ft^3 at 30 s
V = (4/3) pi r^3
so r = 0.71 ft at 30 s
dV/dr = 4 pi r^2 (the surface area of course)
but
dV/dt = dV/dr * dr/dt
so
dr/dt = dV/dt / (4 pi r^2)
dr/dt = .05 /(4 pi*.71^2)
= .00789 feet/second
aren't you supposed to plug everything in AFTER you derive it?
LOL, either way
I do physics, I find t easier to visualize step by step.
To find the rate of change of the radius, we need to use the information about the rate of change of the volume of the balloon.
Let's start by finding the formula for the volume of a balloon. The volume of a balloon can be calculated using the formula V = (4/3) * π * r^3, where V is the volume and r is the radius.
Now, let's differentiate both sides of the equation with respect to time (t) to find the rate of change of the volume:
dV/dt = d/dt [(4/3) * π * r^3]
dV/dt = (4/3) * π * d/dt(r^3)
To find dV/dt, we can use the information given in the problem: the rate of change of the volume is 3 cubic ft per min. So, let's substitute that value:
3 = (4/3) * π * d/dt(r^3)
Now, we want to find the rate of change of the radius, which is d/dt(r). To solve for that, we can rearrange the equation:
d/dt(r^3) = (3 * 3) / ((4/3) * π)
d/dt(r^3) = 9 / ((4/3) * π)
d/dt(r^3) = (27/4π)
Now we differentiate both sides of the equation with respect to time (t):
d/dt(d/dt(r^3)) = d/dt(27/4π)
Let's simplify the right side of the equation:
0 = 0
So, the rate of change of the radius is 0. This means that after 30 seconds, the radius is not changing.