Air is being pumped into a spherical hot air balloon at a rate of 50 cm^3/min. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 200 cm.
V = (4/3)πr^3
dV/dt = 4π r^2 dr/dt
plug in the given stuff
50 = 4π(100)^2 dr/dt
dr/dt = 50/(4π(100)^2 ) = .... cm/min
3.97cm/min
To determine the rate at which the radius of the balloon is increasing, we need to use related rates and the formula for the volume of a sphere.
First, let's set up the problem.
Given:
- Air is being pumped into the balloon at a rate of 50 cm^3/min
- The diameter of the balloon is 200 cm
We want to find:
- The rate at which the radius of the balloon is increasing
Let's define some variables:
- V: the volume of the balloon (in cm^3)
- r: the radius of the balloon (in cm)
- t: time (in min)
Now, let's analyze the problem and set up the equation relating the variables.
The volume of a sphere can be expressed as V = (4/3) * π * r^3.
The rate at which the volume of the balloon is changing with time is given as dV/dt = 50 cm^3/min.
Since the diameter of the balloon is 200 cm, we can find the radius using the formula r = d/2, where d is the diameter.
Substituting this into the volume equation, we have V = (4/3) * π * (d/2)^3 = (4/3) * π * (d^3/8) = (1/6) * π * d^3.
Now, let's differentiate both sides of the equation with respect to time t.
dV/dt = (1/2) * π * 3 * d^2 * (dd/dt).
Since we are looking for the rate at which the radius (r) is changing, we can relate d and r using the equation r = d/2. Thus, dd/dt = 2 * dr/dt.
Now, we can substitute the given values and solve for dr/dt, the rate at which the radius of the balloon is increasing.
dV/dt = (1/2) * π * 3 * (200)^2 * (2 * dr/dt).
Simplifying this equation, we have:
50 = 600 * π * (dr/dt).
Finally, we can solve for dr/dt, the rate at which the radius of the balloon is increasing:
dr/dt = 50 / (600 * π).
Now, we can calculate the value:
dr/dt ≈ 0.026 cm/min.
Therefore, when the diameter of the balloon is 200 cm, the rate at which the radius of the balloon is increasing is approximately 0.026 cm/min.