To determine the time it would take for the balloon to inflate, we need to calculate the volume of the balloon and then divide it by the rate of inflation.
The circumference of the balloon is given as 16 inches. We can use this information to find the radius (r) of the balloon by using the formula:
Circumference = 2πr
16 = 2πr
Dividing both sides by 2π:
r = 16 / (2π)
Now, we can calculate the volume (V) of the balloon using the formula for the volume of a sphere:
V = (4/3)πr^3
Substituting the value of r we found earlier:
V = (4/3)π * (16 / (2π))^3
V ≈ (4/3) * (16 / (2π))^3
V ≈ (4/3) * (8 / π)^3
V ≈ (4/3) * (512 / π^3)
V ≈ (4 * 512) / (3 * π^3)
V ≈ 2048 / (3 * π^3)
Now, we can divide the volume (2048 / (3 * π^3)) by the inflation rate of 65 cubic inches per second to find the time it takes to inflate the balloon:
Time = Volume / Rate
Time = (2048 / (3 * π^3)) / 65
Time ≈ (2048 / (3 * 3.14^3)) / 65
Time ≈ (2048 / (3 * 31.0064)) / 65
Time ≈ (2048 / 93.0192) / 65
Time ≈ 0.02204 / 65
Time ≈ 0.000338 hours
Converting hours to seconds:
Time ≈ 0.000338 hours * 3600 seconds/hour
Time ≈ 1.2168 seconds
Therefore, it would take approximately 1.22 seconds for the balloon to inflate at a rate of 65 cubic inches per second.