A spherical balloon is inflated so that its volume is increasing at the rate of 3.2 ft3/min. How rapidly is the diameter of the balloon increasing when the diameter is 1.2 feet?
To find the rate at which the diameter of the balloon is increasing, we need to use the relationship between the volume and the diameter of a sphere.
The volume of a sphere can be given by the formula:
V = (4/3)πr^3
Since we need to find the rate of change of the diameter, we will express the volume in terms of the diameter instead of the radius. The relationship between the radius (r) and the diameter (d) is:
r = d/2
Let's differentiate the volume equation with respect to time (t) using implicit differentiation:
dV/dt = 4/3 * π * (3r^2 * dr/dt)
Where dV/dt is the rate of change of the volume and dr/dt is the rate of change of the radius.
Now, let's substitute the given values into the equation:
dV/dt = 3.2 ft^3/min
We are given that the diameter (d) is 1.2 feet, so we can find the radius (r):
r = d/2 = 1.2/2 = 0.6 feet
Now we can solve for dr/dt, the rate at which the radius is changing:
3.2 = 4/3 * π * (3 * (0.6)^2 * dr/dt)
Simplifying the equation:
3.2 = 0.8π * (0.6^2 * dr/dt)
dr/dt = 3.2 / (0.8π * 0.36)
dr/dt = 3.2 / (0.288π)
dr/dt ≈ 3.51 ft/min
Therefore, the diameter of the balloon is increasing at a rate of approximately 3.51 feet per minute when the diameter is 1.2 feet.
To find the rate at which the diameter of the balloon is increasing, we need to first establish a relationship between the volume and the diameter.
The volume of a sphere can be calculated using the formula: V = (4/3)πr³, where V is the volume and r is the radius of the sphere.
Since the diameter is twice the radius (d = 2r), we can rewrite the formula as: V = (4/3)π(d/2)³.
To find the rate at which the diameter is increasing (dd/dt), we need to find the derivative of the volume formula with respect to time (dV/dt).
Differentiating the volume formula with respect to time (dV/dt), we get:
dV/dt = (4/3)π(3/2)(d/2)²(dd/dt).
Simplifying the equation, we have:
3.2 = (4/3)π(3/2)(1.2/2)²(dd/dt).
Now, we can solve for dd/dt, the rate at which the diameter is increasing:
dd/dt = (3.2 * 3) / [(4/3)π(3/2)(1.2/2)²].
dd/dt = 2.56 / π (1.8) (0.6)².
dd/dt ≈ 0.82 ft/min.
Therefore, the diameter of the balloon is increasing at a rate of approximately 0.82 feet per minute when the diameter is 1.2 feet.
V = (4/3)π r^3
dV/dt = 4π r^2 dr/dt
when r = .6 , dV/dt = 3.2
3.2 = 4π(.6)^2 dr/dt
dr/dt = 3.2/(1.44π) = appr .707
so d(diameter)/dt = 1.415 ft/min