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.
If the design width doubles, then I'm assuming that the surface area of the balloon doubles when the balloon is inflated a second time. And the balloon can be considered as a sphere.
Let's let the original radius = r
Then, the original surface area of the balloon = 4pi *r^2
Then, for the suface area to double, the radius must increase by a factor of √2
Proof : 4pi (√2 r)^2 = 4pi (2)r^2 = 8pi*r^2 = twice the original surface area
So...let's find the original radius using the volume for a sphere
71 =(4/3)pi*r^3 divide both sides by (4/3)pi
71/ [ (4/3)pi] = r^3 take the cube root of both sides
cube root ( 71/ [ (4/3)pi] ) = r = about 2.5687cm^3
So......if the original radius increases by a factor of √2, the new volume is :
V = (4/3)pi (2.5687√2)^3 = about 200.8 cm^3