A spherical balloon with radius r inches has volume V(r)= 4/3πr^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.

Question

anonymous · Answer

$$\frac{ 4 }{ 3} \pi r^3$$

aum · Answer

Find the volume when radius = (r+1).
Find the volume when radius = r.
Find the difference and that will be the function.

aum · Answer

$$
V(r+1) = \frac 43\pi (r+1)^3 \
V(r) =  \frac 43\pi r^3 \
V(r+1) - V(r) =  \frac 43\pi (r+1)^3  - \frac 43\pi r^3 \
=  \frac 43\pi ((r+1)^3 - r^3) =  \frac 43\pi (r+1-r)((r+1)^2 - (r+1)r + r^2) \
=  \frac 43\pi (r^2 + 2r + 1 - r^2 - r + r^2) \
=  \frac 43\pi (r^2 + r + 1)
$$

aum · Answer

small mistake: (a^3 - b^3) = (a-b)(a^2 + ab + b^2).

anonymous · Answer

what i did is just substituted r+1 into r^3. Why is that wrong?

anonymous · Answer

and why are we trying to find the difference?

aum · Answer

If they ask you how much air will be required to fill the balloon to (r+1), then you can just substitute (r+1) for radius in the volume formula and calculate the volume.  But that is not what they are asking.

They are asking the balloon is already at radius r.  How much more air is required to increase the radius to (r+1)?  So you have ti find the difference in volumes.

aum · Answer

$$
V(r+1) - V(r) =  \frac 43\pi (3r^2 + 3r + 1)
$$

anonymous · Answer

Oh, that makes sense. Thank you very much :)

aum · Answer

You are welcome.