Air is being blown into a spherical balloon at the rate of 75 cm3/s. Determine the rate at which the radius of the balloon is increasing when the radius is 28 cm.
To determine the rate at which the radius of the balloon is increasing, we need to use the relationship between the volume and the radius of a sphere.
The volume of a sphere can be calculated using the formula:
V = (4/3) * π * r^3
where V represents the volume of the sphere and r represents the radius.
Now, let's differentiate both sides of the equation with respect to time (t) to find the rate at which the volume is changing with respect to time:
dV/dt = (4/3) * π * 3r^2 * (dr/dt)
In this equation, dV/dt represents the rate at which the volume (V) is changing, and dr/dt represents the rate at which the radius (r) is changing.
Since we know the rate at which the volume is changing (75 cm^3/s) and want to find the rate at which the radius is changing, we can rearrange the equation and solve for dr/dt:
dr/dt = (1 / ((4/3) * π * 3r^2)) * dV/dt
Now, substitute the given values: dV/dt = 75 cm^3/s and r = 28 cm into the equation:
dr/dt = (1 / ((4/3) * π * 3(28)^2)) * 75
Simplifying this calculation will give us the rate at which the radius is increasing.
v = 4/3 pi r^3
dv/dt = 4 pi r^2 dr/dt
now just plug in your numbers and find dr/dt