A sphere is increasing in volume at the rate of 3(pi) cm^3/sec. At what rate is the radius changing when the radius is 1/2 cm? The volume of a sphere is given by V=4/3(pi)r^3.
A. pi cm/sec
B. 3 cm/sec
C. 2 cm/sec
D. 1 cm/sec
E. .5 cm/sec
B. 3 cm/s
from you formula for v,
dv/dt = 4pi r^2 dr/dt
now plug in the numbers.
To find the rate at which the radius is changing, we need to use related rates and differentiate the equation for volume with respect to time.
Given:
Rate of change of volume (dV/dt) = 3(pi) cm^3/sec
Radius (r) when the rate is to be determined = 1/2 cm
The volume of a sphere is given by V = (4/3)(pi)r^3.
Differentiating both sides of the equation with respect to time (t):
dV/dt = d/dt[(4/3)(pi)r^3]
Using the power rule, we can differentiate r^3 as follows:
dV/dt = (4/3)(pi) * d/dt(r^3)
Now we need to find d/dt(r^3), which is the rate of change of the radius with respect to time. Let's call it dr/dt:
dV/dt = (4/3)(pi) * 3r^2 * dr/dt
Substituting the given rate of change of volume and radius:
3(pi) = (4/3)(pi) * 3(1/2)^2 * dr/dt
Simplifying:
3(pi) = (4/3)(pi) * (3/4) * dr/dt
The (pi) cancels out:
3 = (3/4) * dr/dt
Simplifying further:
dr/dt = 3 * (4/3)
dr/dt = 4 cm/sec
Therefore, the rate at which the radius is changing when the radius is 1/2 cm is 4 cm/sec.
The correct answer is option B: 4 cm/sec.
To find the rate at which the radius is changing, we need to use the given information about the rate at which the volume is changing.
We are given that the volume of the sphere is increasing at a rate of 3(pi) cm^3/sec. The volume of a sphere is given by V = (4/3)(pi)r^3.
Now, let's differentiate the equation with respect to time (t) using implicit differentiation:
dV/dt = 4(pi)(r^2)(dr/dt)
Here, dV/dt represents the rate of change of volume (3(pi) cm^3/sec), dr/dt represents the rate at which the radius is changing (what we are looking for), and r represents the radius of the sphere.
Plugging in the given rate of change of volume:
3(pi) = 4(pi)(r^2)(dr/dt)
Now, let's solve for dr/dt:
dr/dt = (3(pi))/(4(pi)(r^2))
Since we want to find the rate at which the radius is changing when the radius is 1/2 cm, we can substitute r = 1/2 into the equation:
dr/dt = (3(pi))/(4(pi)((1/2)^2))
dr/dt = (3(pi))/(4(pi)(1/4))
dr/dt = (3(pi))/(pi)
dr/dt = 3
So, the rate at which the radius is changing when the radius is 1/2 cm is 3 cm/sec.
Therefore, the answer is option B. 3 cm/sec.