Application Problem: Ball Inflation (copy)

Volume of a sphere

The formula is given by V = (4/3)πr³, where r is the radius.

Radius function of time

The function that models the radius of the sphere over time: r(t) = 1.5t + 3.

Volume after 4 seconds

The volume of the sphere after 4 seconds is V = 972π cm³.

Function composition

The process of combining functions; in this context: (f ∘ g)(t) = f(g(t)) = (4/3)π(1.5t + 3)³.

Common pitfall in volume calculation

Forgetting to include the factor of π when calculating the volume.

Units for radius, time, and volume

Radius is measured in cm, time in seconds, and volume in cm³.

Impact of radius changes on volume

As the radius increases linearly, the volume increases as the cube of the radius, leading to significant volume changes with small radius changes.

Starting diameter

The initial diameter of the inflated ball is 6 cm, which corresponds to a starting radius of 3 cm.

Correct volume expression

The correct final expression for volume after 4 seconds is V = 972π cm³, approximately 3053.63 cm³.

Importance of distinguishing diameter and radius

It is crucial to know whether to use radius or diameter in volume formulas.