A spherical balloon is to be deflated so that its radius decreases at a constant rate of 12 cm/min. At what rate must air be removed when the radius is 5 cm? Must be accurate to the 5th decimal place.
I keep getting -3769.91118. What am I doing wrong? Please help!
To find the rate at which air must be removed, you need to use the formula for the volume of a sphere:
V = (4/3)πr^3
Differentiate this equation with respect to time (t) using the chain rule:
dV/dt = dV/dr * dr/dt
Given that dV/dt is the rate at which air is being removed from the balloon and dr/dt is the rate at which the radius is decreasing, we can substitute these values into the equation.
dV/dr represents the derivative of the volume with respect to the radius. So, when you differentiate the equation for the volume of a sphere, you get:
dV/dr = 4πr^2
Now, we know that dr/dt is given as -12 cm/min (since we're looking for the rate of contraction).
At the given radius of r = 5 cm, we can substitute these values into the equation:
dV/dt = (4π(5)^2) * (-12)
Simplifying this expression:
dV/dt = 100π * (-12)
dV/dt = -1200π
Now, to find the numerical value, calculate -1200 * π ≈ -3769.91118.
Hence, the correct answer is -3769.91118 cm^3/min.
So, it seems like you made a mistake when calculating the value of -1200 * π. Make sure you are using an accurate approximation for π, preferably to at least the 5th decimal place, when performing calculations to avoid rounding errors.