$\"\"$

The volume of the spherical balloon with radius $\"r\"$ is $\"v(r)=\\frac{4}{3}\\pi$ .

Find the function that the amount of air to inflate the balloon from a radius $\"r\"$ inches to a radius $\"r+1\"$ inches.

Then the resultant function is subtract $\"v(r+1)-v(r)\"$ .

$\"v(r+1)-v(r)=\\frac{4}{3}\\pi$

Apply formula : $\"(a+b)^3=a^3+3a^2b+3ab^2+b^3\"$ .

$\"=\\frac{4}{3}\\pi$

$\"=\\frac{4}{3}$

Take out 3 as a common factor.

$\"=\\frac{4}{3}$

$\"=4\\pi$

$\"\"$

The resultant function is $\"4\\pi$ .