a spherical balloon with radius r inches has volume V9r) =(4/3)πr^3. Find a function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r+1 inches

Additional volume of air required

=f(r+1)-f(r)
=(4/3)π(r+1)&sup3 - (4/3)πr³
= ...

To find the amount of air required to inflate the balloon from a radius of r inches to a radius of r+1 inches, we need to calculate the difference in volume between the two radii.

The formula for the volume of a sphere is V = (4/3)πr^3, where V is the volume and r is the radius.

Let's calculate the volume at r+1 inches:
V(r+1) = (4/3)π(r+1)^3

Now, let's subtract the volume at r inches from the volume at r+1 inches to find the difference:
V(r+1) - V(r) = (4/3)π(r+1)^3 - (4/3)πr^3

Simplifying the equation:
V(r+1) - V(r) = (4/3)π(r^3 + 3r^2 + 3r + 1) - (4/3)πr^3
V(r+1) - V(r) = (4/3)π(r^3 + 3r^2 + 3r + 1 - r^3)

Now, let's simplify further:
V(r+1) - V(r) = (4/3)π(3r^2 + 3r + 1)

Therefore, the function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r+1 inches is:
f(r) = (4/3)π(3r^2 + 3r + 1)