Let's label the volume of air that goes into the balloon from one full breath B. With this information, we can illustrate the two states of the balloon including its volume and width.
At the full width the balloon is twice as wide as the balloon when it is at half of the width. This means the linear scale factor between these two states is 2.
linear scale factor=2
If we cube both sides of this equation, we can calculate the corresponding volume scale factor when the linear scale factor is 2.
(Linear scale factor)^3=( 2)^3
⇓
Volume scale factor=8
The big balloon has a volume that is 8 times greater than the small balloon. We know that each breath puts a volume of B into the balloon. Therefore, after 3 breaths the volume of the balloon is 3B. If we multiply this by 8, we get the volume of the big balloon.
3B(8)=24B
The big balloon has a volume of 24 full breaths. Since the small balloon already contains 3 breaths, Koy needs 21 more breaths to fill the balloon.
Alternative Solution
Calculate volumes
A balloon has the shape of a sphere. The volume of a sphere is the product of π, the radius cubed, and 43. We know that after three full breaths the balloon is half of the width she needs. If we call the full width she needs 2r and the volume of air from one breath B, then we can write the following equation to describe the volume and width after the three first breaths.
3B=4Ď€ r^3/3
If we solve for B we can determine the volume of one breath.
Each breath puts a volume of 4Ď€ r^39 into the balloon. If she needs x breaths to fill the balloon to the desired width, we can write the following equation.
(4Ď€ r^3/9)x = 4Ď€ (2r)^3/3
Let's solve this equation for x
In total she needs to put 24 full breaths into the balloon to fill it to the width that she needs. Since she has already taken 3 breaths, she has 21 full breaths left.
