A design on the surface of a balloon is 5 cm wide when the balloon holds 71 cm^3 of air. How much air does the balloon hold when the design is 10 cm wide? Explain the method you use to find the amount of air.
since the linear scale has doubled, the volume has grown by a factor of 8.
To find the amount of air the balloon holds when the design is 10 cm wide, we can use the concept of ratios and proportions.
We know that when the design is 5 cm wide, the balloon holds 71 cm^3 of air. Let's call this volume V1. Now, let's say when the design is 10 cm wide, the balloon holds a volume V2, which is what we want to find.
We can set up a proportion to relate the two widths and volumes:
5 cm / 10 cm = V1 / V2
Simplifying this proportion, we get:
1/2 = V1/V2
Now, we know V1 is 71 cm^3. Substituting this into the equation, we have:
1/2 = 71 cm^3 / V2
To solve for V2, we can cross-multiply:
2 * 71 cm^3 = V2
V2 = 142 cm^3
Therefore, when the design is 10 cm wide, the balloon holds 142 cm^3 of air.
The method used here is based on the concept of ratios and proportions. By setting up a proportion, we can relate the given width and volume to the unknown width and volume, and solve for the unknown value algebraically.