A spherical balloon with a radius r inches has volume V(r)= 4/3 Pi r^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.
An explanation would be appreciated, as I have no clue how to even begin.
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 volume of the balloon with radius r is given by V(r) = (4/3)πr^3.
Similarly, the volume of the balloon with radius r+1 is given by V(r+1) = (4/3)π(r+1)^3.
To find the amount of air required to inflate the balloon, we need to find the difference in volume between these two cases, which can be calculated as:
Amount of air required = V(r+1) - V(r)
= (4/3)π(r+1)^3 - (4/3)πr^3
We can simplify this expression further:
Amount of air required = (4/3)π(r+1)^3 - (4/3)πr^3
= (4/3)π[(r+1)^3 - r^3]
Expanding the bracket (r+1)^3, we have:
Amount of air required = (4/3)π(r^3 + 3r^2 + 3r + 1 - r^3)
= (4/3)π(3r^2 + 3r + 1)
So, 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)
To 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, we need to find the difference in volumes between the two radii.
The volume of the balloon with radius r inches is given by V(r) = (4/3)πr^3.
The volume of the balloon with radius r + 1 inches is given by V(r + 1) = (4/3)π(r + 1)^3.
To find the difference in volumes, we subtract the initial volume from the final volume:
V(r + 1) - V(r) = (4/3)π(r + 1)^3 - (4/3)πr^3
= (4/3)π[(r + 1)^3 - r^3]
= (4/3)π[(r^3 + 3r^2 + 3r + 1) - r^3] (Expanding (r + 1)^3)
Simplifying 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 given by:
F(r) = (4/3)π[3r^2 + 3r + 1]
This function F(r) gives the difference in volume, which represents the amount of air required.