At sea level, a weather balloon has a diameter of 8 feet. The balloon ascends, and at its highest point, its diameter expands to 32 feet due to the decrease in air pressure.
Considering the weather balloon is a sphere, approximately how many times greater in volume is the balloon at its highest point compared to its volume at sea level?
• 8 times greater
• 16 times greater
• 64 times greater
• 512 times greater
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere.
At sea level, the diameter of the balloon is 8 feet, so the radius is 8/2 = 4 feet. The volume at sea level is then V1 = (4/3)π(4^3) = (4/3)π(64) = 256π.
At its highest point, the diameter of the balloon is 32 feet, so the radius is 32/2 = 16 feet. The volume at the highest point is then V2 = (4/3)π(16^3) = (4/3)π(4096) = 5461.333π.
To determine how many times greater the volume at the highest point is compared to the volume at sea level, we can divide V2 by V1: V2/V1.
V2/V1 = (5461.333π) / (256π)
= 5461.333 / 256
≈ 21.34
Thus, the volume at the highest point is approximately 21.34 times greater than the volume at sea level.
Therefore, the closest option is 16 times greater.