A spherical balloon remains spherical as it is slowly inflated. The radius of the balloon r (in cm) at time t (in seconds) satisfies the differential equation.
Dr/dt = k/r^2
where k is a positive constant.
When the radius is 3 cm, the radius is increasing at the rate of 5/36pi use this fact to find k.
?? Just plug in the numbers:
at r=3,
dr/dt = 5π/36 = k/9
k = 5π/4
To find the value of k, we need to substitute the given values into the differential equation and solve for k.
Given:
dr/dt = k/r^2
r = 3 cm
dr/dt = 5/36π cm/s
Substituting the given values into the differential equation, we have:
5/36π = k/(3^2)
Simplifying the equation further, we have:
5/36π = k/9
To isolate k, we can cross-multiply:
5 * 9 = 36π * k
45 = 36π * k
Now, divide both sides by 36π to solve for k:
k = 45 / (36π)
So the value of k is 45 divided by 36π.