A spherical balloon with radius r inches has volume V(r) = 43𝜋r3. Find an expression that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 2 inches. (Express your answer in terms of 𝜋 and r.)

The volume of the balloon with radius r is given by V(r) = 4/3πr³.

The volume of the balloon with radius r + 2 is given by V(r + 2) = 4/3π(r + 2)³.

The amount of air required to inflate the balloon from a radius of r inches to a radius of r + 2 inches is the difference between these two volumes.

So, the amount of air required is V(r + 2) - V(r) = 4/3π(r + 2)³ - 4/3πr³.

Simplify the expression to get the final answer:

= 4/3π[(r + 2)³ - r³] = 4/3π[r³ + 6r² + 12r + 8 - r³] = 4/3π[6r² + 12r + 8]

So, the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 2 inches is 4/3π[6r² + 12r + 8] cubic inches.

The volume of the balloon with radius r is given by V(r) = 4/3πr³.

The volume of the balloon with radius r + 2 is given by V(r + 2) = 4/3π(r + 2)³.

The amount of air required to inflate the balloon from a radius of r inches to a radius of r + 2 inches is the difference between these two volumes.

So, the amount of air required is V(r + 2) - V(r) = 4/3π(r + 2)³ - 4/3πr³.

Simplify the expression to get the final answer:

= 4/3π[(r + 2)³ - r³] = 4/3π[r³ + 6r² + 12r + 8 - r³] = 4/3π[6r² + 12r + 8] = 8πr² + 16πr + 32/3π.

So, the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 2 inches is 8πr² + 16πr + 32/3π cubic inches.