A spherical balloon with a radius "r" inches has volume V(r)=(4/3)(pi)(r^3). Find 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.
Plug in r+1 where "r" is in the V(r) formula. Then subtract V(r)= (4/3) pi r^3
You will get a new formula for the increase in volume
To find 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, we need to find the difference in volume between the two radii.
Let's start by finding the volume of the balloon with a radius of "r+1" inches. We can use the volume formula V(r)=(4/3)(pi)(r^3) for this.
V(r+1) = (4/3)(pi)((r+1)^3)
V(r+1) = (4/3)(pi)(r^3 + 3r^2 + 3r + 1)
Next, let's find the volume of the balloon with a radius of "r" inches using the same formula.
V(r) = (4/3)(pi)(r^3)
Now, 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 subtract the two volumes:
Amount of air required = V(r+1) - V(r)
Amount of air required = [(4/3)(pi)(r^3 + 3r^2 + 3r + 1)] - [(4/3)(pi)(r^3)]
Amount of air required = (4/3)(pi)(r^3 + 3r^2 + 3r + 1 - r^3)
Amount of air required = (4/3)(pi)(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)(pi)(3r^2 + 3r + 1)