Question

A spherical balloon is inflated with gas at a rate of 600 cubic centimeters per minute.

(a) Find the rates of change of the radius when r = 50 centimeters and r = 85 centimeters.
r = 50 ? cm/min
r = 85 ? cm/min

(b) Explain why the rate of change of the radius of the sphere is not constant even though dv/dt is constant.

A.) dr/dt as a function runs parallel to the volume function, which is not linear

B.) The rate of change of the radius is a linear relationship whose slope is dV/dt

C.) The rate of change of the radius is a cubic relationship.

D.) The volume only appears constant; it is actually a rational relationship.

E.) dr/dt depends on r2, not simply r.

Answer 1

To find the rates of change of the radius of a spherical balloon, we can use the formula V = 4/3πr^3 for the volume of a sphere and take the derivative of both sides with respect to time. The rate of change of the radius is not constant even though the volume is changing at a constant rate because the relationship between the radius and the volume of a sphere is not linear. The rate of change of the radius depends on r^2, not simply r.

Step-by-step explanation:

To find the rates of change of the radius, we can use the formula V = 4/3πr^3 for the volume of a sphere. Taking the derivative of both sides with respect to time gives us dV/dt = 4πr^2(dr/dt), where dV/dt is the rate at which the volume is changing and dr/dt is the rate at which the radius is changing.

(a) When r = 50 cm, we know that dV/dt = 600 cm^3/min. Plugging in these values into the equation gives us 600 = 4π(50)^2(dr/dt). Solving for dr/dt, we find that dr/dt = 600 / (4π(50)^2).

Similarly, when r = 85 cm, we can plug in the values into the equation to find the rate of change of the radius.

(b) The rate of change of the radius is not constant even though the volume is changing at a constant rate because the relationship between the radius and the volume of a sphere is not linear. The volume of a sphere is proportional to the cube of the radius, so small changes in the radius can result in larger changes in the volume. Hence, the rate of change of the radius depends on r^2, not simply r.