The rate of change of the radius of a sphere is constant. The rate of change of the volume is?
a. Increasing at any time
b. Decreasing at any time
c. Increasing when dr/dt > 0 and decreasing when dr/dt < 0
d. Decreasing when dr/dt > 0 and increasing when dr/dt < 0
The rate of change of the volume of a sphere can be determined using the formula for the volume of a sphere, V = (4/3)πr^3, where V represents the volume and r represents the radius.
To find the rate of change of the volume, we need to take the derivative of the volume formula with respect to time (t).
dV/dt = d/dt[(4/3)πr^3]
Using the chain rule, we can rewrite this as:
dV/dt = (4/3)π * d(r^3)/dt
To find d(r^3)/dt, we can use the power rule for differentiation, which states that d(x^n)/dt = nx^(n-1) * dx/dt.
In this case, we have n = 3 and x = r, so:
d(r^3)/dt = 3r^2 * dr/dt
Substituting this back into the equation for dV/dt, we get:
dV/dt = (4/3)π * 3r^2 * dr/dt
Simplifying, we have:
dV/dt = 4πr^2 * dr/dt
From this equation, we can see that the rate of change of the volume depends on both the radius of the sphere (r) and the rate of change of the radius (dr/dt).
Therefore, the correct answer is option c: Increasing when dr/dt > 0 and decreasing when dr/dt < 0.
To determine the rate of change of the volume of a sphere, we need to understand the relationship between the radius and the volume.
The volume of a sphere is given by the formula V = (4/3)πr^3, where V represents the volume and r represents the radius.
Given that the rate of change of the radius (dr/dt) is constant, we can differentiate the volume formula with respect to time (dt) to find the rate of change of the volume (dV/dt).
Differentiating the volume formula, we get:
dV/dt = d/dt[(4/3)πr^3]
= (4/3)π * d/dt(r^3) [Applying the power rule of differentiation]
= (4/3)π * 3r^2 * dr/dt [Applying the chain rule]
= 4πr^2 * dr/dt
From this expression, we can see that the rate of change of the volume (dV/dt) is directly proportional to the square of the radius (r^2) and the rate of change of the radius (dr/dt).
Now let's consider the options:
a. Increasing at any time: This option is not necessarily true, as the rate of change of the volume depends on the rate of change of the radius.
b. Decreasing at any time: This option is not necessarily true, as the rate of change of the volume depends on the rate of change of the radius.
c. Increasing when dr/dt > 0 and decreasing when dr/dt < 0: This option is correct. Since the rate of change of the volume is proportional to the rate of change of the radius (dr/dt), the volume will increase when the radius is increasing (dr/dt > 0) and decrease when the radius is decreasing (dr/dt < 0).
d. Decreasing when dr/dt > 0 and increasing when dr/dt < 0: This option is incorrect. The rate of change of the volume is directly proportional to the square of the radius (r^2) and the rate of change of the radius (dr/dt). Therefore, the rate of change of the volume will increase when the radius is increasing (dr/dt > 0) and decrease when the radius is decreasing (dr/dt < 0).
So, the correct answer is option c. The rate of change of the volume is increasing when dr/dt > 0 and decreasing when dr/dt < 0.
dv/dt = 4πr^2 dr/dt
so, what do you think?
Technically, none of the choices is correct, but that's only because you stated the question incorrectly.