Air is pumped into a spherical balloon at the rate of 20cm^3/s^-1.find the rate of change of radius of the balloon when the radius is 6cm
V = (4/3)π r^3
dV/dt = 4π r^2 dr/dt
20 = 4π (36) dr/dt
dr/dt = 20/(144π) cm/s
= 5/(36π) cm/s
To find the rate of change of the radius, we need to use the volume formula for a sphere.
The volume of a sphere is given by the formula:
V = (4/3) * pi * r^3
Where V is the volume and r is the radius.
We are given that the rate of change of volume is 20 cm^3/s, so we can differentiate the volume equation with respect to time (t) to find the derivative:
dV/dt = 4 * pi * r^2 * dr/dt
Here, dr/dt represents the rate of change of the radius.
We are also given that the radius (r) is 6 cm. Plugging in these values, we have:
20 = 4 * pi * 6^2 * dr/dt
Now, rearrange the equation to solve for dr/dt:
dr/dt = 20 / (4 * pi * 6^2)
Calculating this equation, we find:
dr/dt ≈ 0.00559 cm/s
Therefore, the rate of change of the radius of the balloon when the radius is 6 cm is approximately 0.00559 cm/s.
To find the rate of change of the radius of the balloon, we need to use the related rates method in calculus.
Let's start by assigning some variables:
- Let r be the radius of the balloon (in cm).
- Let V be the volume of the balloon (in cm^3).
- Let t be the time (in s).
We are given that air is pumped into the spherical balloon at the rate of 20 cm^3/s. This implies that the rate of change of the volume of the balloon with respect to time is given by dV/dt = 20 cm^3/s.
The volume of a sphere is given by V = (4/3) * π * r^3. We can differentiate this equation with respect to time to get dV/dt:
dV/dt = 4πr^2(dr/dt).
Since we are interested in finding the rate of change of the radius (dr/dt), we need to rearrange the equation:
dr/dt = (dV/dt) / (4πr^2).
Now, we can substitute the given values into the equation. When the radius is 6 cm, we have:
r = 6 cm,
dV/dt = 20 cm^3/s.
Plugging these values into the equation, we get:
dr/dt = 20 cm^3/s / (4π(6 cm)^2).
Simplifying this expression:
dr/dt = 20 cm^3/s / (4π(36 cm^2)).
≈ 20 cm^3/s / (144π cm^2).
≈ 0.044 cm/s.
So, the rate of change of the radius of the balloon when the radius is 6 cm is approximately 0.044 cm/s.