A balloon is being filled with air at a rate of 3 cubic meters per minute. Find the rate at which the radius is changing when the volume is 113.0973355 cubic meters.
Well, that's certainly an inflated situation! Let's calculate the rate at which the radius is changing.
We know that the volume V of a balloon can be calculated using the formula V = (4/3)πr³, where r is the radius.
Now, we can take the derivative of both sides of the equation with respect to time t, and use the chain rule to find the rate at which the volume is changing with respect to time: dV/dt = (4π/3)(3r²)(dr/dt).
Given that dV/dt = 3 cubic meters per minute, we can substitute this value into the equation, along with the value for V, which is 113.0973355 cubic meters. Solving for dr/dt will give us the rate at which the radius is changing:
3 = (4π/3)(r²)(dr/dt)
dr/dt = 3 / [(4π/3)(r²)]
dr/dt = 9 / (4πr²)
So, when the volume is 113.0973355 cubic meters, the rate at which the radius is changing is equal to 9 divided by 4πr².
To find the rate at which the radius is changing, we can use related rates by connecting the formula for the volume of a sphere to the radius.
The formula for the volume of a sphere is V = (4/3) * π * r^3, where V is the volume and r is the radius.
Taking the derivative with respect to time, we get dV/dt = 4 * π * r^2 * dr/dt, where dV/dt represents the rate at which the volume changes over time and dr/dt represents the rate at which the radius changes over time.
Given that the rate at which the volume is changing is 3 cubic meters per minute (dV/dt = 3) and the volume is 113.0973355 cubic meters (V = 113.0973355), we can substitute these values into the equation:
3 = 4 * π * r^2 * dr/dt.
We can rearrange this equation to solve for dr/dt:
dr/dt = 3 / (4 * π * r^2).
To find the rate at which the radius is changing, we need to know the radius of the balloon. Without this information, we cannot calculate the exact value for dr/dt.
To find the rate at which the radius of the balloon is changing, we need to use the relationship between the volume, radius, and time. We can start by expressing the volume of the balloon as a function of the radius.
We know that the volume of a sphere is given by the formula:
V = (4/3) * π * r^3
Let's differentiate this equation implicitly with respect to time (t) to express the rate of change of volume (dV/dt) in terms of the rate of change of radius (dr/dt):
(dV/dt) = (dV/dr) * (dr/dt)
Now, let's substitute the given values:
dV/dt = 3 m^3/min
V = 113.0973355 m^3
To find (dV/dr), we can differentiate the volume equation with respect to the radius (r):
(dV/dr) = 4π * r^2
Substituting all the known values, we have:
3 = (4π * r^2) * (dr/dt)
Now, we can solve this equation to find the rate at which the radius is changing (dr/dt) when the volume is 113.0973355 cubic meters.
1. Rearrange the equation: (dr/dt) = 3 / (4π * r^2)
2. Substitute the given volume: r = (3V / 4π)^(1/3) = (3 * 113.0973355 / (4π))^(1/3)
3. Calculate the value of r.
4. Substitute the value of r into the equation (dr/dt) = 3 / (4π * r^2).
5. Calculate the value of (dr/dt) to find the rate at which the radius is changing.
By following these steps, you can find the rate at which the radius is changing when the volume is 113.0973355 cubic meters.
V = (4/3)πr^3
dV/dt = 4πr^2 dr/dt
we know dV/dt, but we have to find r
113.0973355 = (4/3)πr^3 , ( I am sure they gave you that number as 36π)
r^3 = 27
r = 3
subbing in
3 = 4π9) dr/dt
dr/dt = 1/(3π) m/min