The volume of a spherical balloon of radius r cm is V cm3, where .
Find dv/dr
My answer: 4πr^2
The balloon leaks at 5cm^3 per second, how fast does the radius of the balloon leaks when the radius is 15 cm.
Find the rate at which the radius is changing with time when the volume is 680 cm^3.
yes, dV/dr = 4 pi r^2 which is in fact the surface area (note, the surface area * dr = change in volume for small dr)
You already know that dV/dr = 4 pi r^2
so calculate dV/dr
then
dV/dt = dV/dr * dr/dt
-5 = dV/dr * dr/dt
so calculate dr/dt from that
part 2
(4/3) pi r^3 = 680
so calculate new r and dr/dt
To find dv/dr, we can use the formula for the volume of a sphere, which is V = (4/3)πr^3.
Taking the derivative of both sides with respect to r, we get:
dV/dr = 4πr^2
So, your answer is correct. dv/dr = 4πr^2.
Now, let's move on to the second part of your question.
To find the rate at which the radius is changing when the volume is 680 cm^3, we need to use related rates.
Given that the balloon leaks at a rate of 5 cm^3 per second, we can use the equation V = (4/3)πr^3 to relate the change in volume to the change in radius.
Differentiating both sides of the equation with respect to time (t), we get:
dV/dt = (4π/3)(3r^2)(dr/dt)
Since we are interested in finding the rate at which the radius is changing with time (dr/dt), we can rearrange the equation to solve for it:
dr/dt = (dV/dt) / [(4π/3)(3r^2)]
To find the rate at which the radius is changing when the radius is 15 cm, we substitute the given values into the equation:
dr/dt = (dV/dt) / [(4π/3)(3(15)^2)]
Next, we also need to know the value of dV/dt (the rate at which the volume is changing), which we can assume is -5 cm^3/s because the balloon is leaking at a rate of 5 cm^3 per second.
Substituting this value into the equation, we get:
dr/dt = (-5) / [(4π/3)(3(15)^2)]
By calculating this expression, we can find the rate at which the radius is changing with time when the radius is 15 cm.
Similarly, to find the rate at which the radius is changing with time when the volume is 680 cm^3, we substitute V = 680 cm^3 into the equation:
dr/dt = (dV/dt) / [(4π/3)(3r^2)]
Since we do not have the value of dV/dt for this case, we are unable to find the rate at which the radius is changing. Additional information is required in order to solve for dr/dt in this situation.