Helium is pumped into a spherical balloon at a rate of 4 cubic feet per second. How fast is the radius increasing after 2 minutes?
Rate of change of radius (in feet per second) = ??
(i dont know what to with the time given to solve)
What's a balloon's favorite type of music? Pop!
But let's get back to solving your question. We are given that helium is being pumped into the balloon at a rate of 4 cubic feet per second. This means that the volume of the balloon is increasing at a rate of 4 cubic feet per second.
We can use the formula for the volume of a sphere to relate the rate of change of the radius with the rate of change of the volume. The formula for the volume of a sphere is V = (4/3)πr^3, where V is the volume and r is the radius.
Differentiating both sides of the equation with respect to time, we get:
dV/dt = (4π/3)(3r^2)(dr/dt)
We are given that dV/dt (the rate of change of volume) is 4 cubic feet per second, so we can plug in these values:
4 = (4π/3)(3r^2)(dr/dt)
Simplifying:
4 = 4πr^2(dr/dt)
Dividing both sides by 4πr^2:
1 = (dr/dt)
So, after 2 minutes, the rate of change of the radius will be 1 foot per second.
Remember, this assumes that the rate of helium being pumped into the balloon remains constant. If the rate changes during the 2 minutes, the result might be different. Also, don't take my balloon jokes too seriously, I'm just clowning around!
To solve this problem, we can use the relationship between the volume of a spherical balloon and its radius. The volume of a sphere is given by the formula V = (4/3)πr^3, where V is the volume and r is the radius.
Given that the helium is pumped into the balloon at a rate of 4 cubic feet per second, we can express this as dV/dt = 4, where dV/dt represents the rate of change of volume with respect to time.
To find the rate at which the radius is increasing, we need to determine the rate of change of the volume with respect to the radius, which can be expressed as dV/dr. By differentiating the volume formula with respect to r, we get dV/dr = 4πr^2.
Now, let's proceed to find the rate of change of the radius after 2 minutes (or 120 seconds).
Since we know that dV/dt = 4, we can substitute this into the expression for dV/dr: 4 = 4πr^2.
Simplifying the equation, we have r^2 = 1/π.
Then, we can solve for the radius r: r = √(1/π).
Now, we can find the rate at which the radius is increasing by differentiating r with respect to t and substituting t = 120 seconds:
dr/dt = d(√(1/π))/dt = (1/2) * (1/√(1/π)) * d(1/π)/dt.
Since d(1/π)/dt = 0 (as π is a constant), the rate at which the radius is increasing is:
dr/dt = (1/2) * (1/√(1/π)) * 0 = 0.
Therefore, the radius is not changing (i.e., it is not increasing) after 2 minutes.
So, the rate at which the radius is increasing after 2 minutes is 0 feet per second.
To solve this problem, we can use the formula for the volume of a sphere:
V = (4/3) * π * r^3,
where V is the volume and r is the radius of the sphere.
Given that helium is pumped into the balloon at a rate of 4 cubic feet per second, we can write:
dV/dt = 4,
where dV/dt represents the rate of change of volume with respect to time.
We want to find the rate at which the radius is increasing, which is dr/dt. To do that, we can differentiate the equation for the volume with respect to time:
d/dt[(4/3) * π * r^3] = dV/dt,
or
(4/3) * π * d(r^3)/dt = dV/dt.
Next, differentiate r^3 with respect to t:
(r^3)' = 3r^2 * (dr/dt).
Substituting this back into the equation, we have:
(4/3) * π * 3r^2 * (dr/dt) = 4.
Simplifying, we get:
4πr^2 * (dr/dt) = 4.
Now, let's solve for dr/dt by rearranging the equation:
dr/dt = 4 / (4πr^2).
Given that we need to find the rate of change of the radius after 2 minutes, we need to convert the time to seconds. Since there are 60 seconds in a minute, 2 minutes is:
2 minutes * 60 seconds/minute = 120 seconds.
Finally, substitute the value into the equation:
dr/dt = 4 / (4πr^2),
dr/dt = 1 / (πr^2),
Let's calculate the value using the given time and any radius value you want to use.
v = 4/3 πr^3
dv/dt = 4πr^2 dr/dt
At t = 120 seconds, v = 480 ft^3
Now you can find r, and you have dv/dt, so use that to find dr/dt.