A spherical hot-air balloon has a diameter of 55 feet. When the balloon is inflated, the radius increases at a rate of 1.5 feet per minute. Approximately how long does it take to inflate the balloon to $\frac{2}{3}$ of its maximum volume? Use $\pi$ =3.14 and $V=\frac{4}{3} \pi r^3$.
A. 16 minutes
B. 18 minutes
C. 23 minutes
D. 26 minutes
Answer (1)
Calculate the maximum radius: r ma x = 2 55 = 27.5 feet.
Determine the radius at 3 2 of the maximum volume: r = 3 3 2 ⋅ 27.5 .
Use the rate of radius increase to find the time: t = 1.5 r .
Approximate the time: t ≈ 16 minutes. 16
Explanation
Problem Setup We are given a spherical hot-air balloon with a diameter of 55 feet. The radius increases at a rate of 1.5 feet per minute. We want to find the time it takes to inflate the balloon to 3 2 of its maximum volume. The volume of a sphere is given by V = 3 4 π r 3 , and we are told to use π = 3.14 .
Maximum Radius First, we find the maximum radius of the balloon, which is half of the diameter: r ma x = 2 55 = 27.5 feet.
Maximum Volume Next, we find the maximum volume of the balloon: V ma x = 3 4 π r ma x 3 = 3 4 ( 3.14 ) ( 27.5 ) 3 .
Target Volume We want to find the volume that is 3 2 of the maximum volume: V = 3 2 V ma x = 3 2 ⋅ 3 4 ( 3.14 ) ( 27.5 ) 3 .
Radius at Target Volume Now, we find the radius r when the volume is 3 2 of the maximum volume. We set V = 3 4 π r 3 and solve for r : 3 4 ( 3.14 ) r 3 = 3 2 ⋅ 3 4 ( 3.14 ) ( 27.5 ) 3 r 3 = 3 2 ( 27.5 ) 3 r = 3 3 2 ⋅ 27.5
Time to Reach Target Radius We are given that the radius increases at a rate of 1.5 feet per minute, so d t d r = 1.5 . We want to find the time t it takes for the radius to reach r from an initial radius of 0. Since the rate is constant, we have r = 1.5 t , so t = 1.5 r .
Calculating Time Substitute the value of r we found in step 5 into the equation t = 1.5 r to find the time t : t = 1.5 3 3 2 ⋅ 27.5 t ≈ 1.5 0.8736 ⋅ 27.5 ≈ 1.5 24.024 ≈ 16.016
Final Answer Therefore, it takes approximately 16 minutes to inflate the balloon to 3 2 of its maximum volume.
Conclusion The time it takes to inflate the balloon to 3 2 of its maximum volume is approximately 16 minutes.
Examples
Understanding rates of change and volume calculations is crucial in many real-world applications. For example, when designing inflatable structures like emergency shelters or recreational equipment, engineers need to accurately predict inflation times and material requirements. This problem demonstrates how calculus concepts can be applied to ensure efficient and safe designs, optimizing both performance and resource utilization.
