A spherical hot-air balloon has a diameter of 55 feet. When the balloon is inflated, the radius increases at a rate of 5 feet per minute. Approximately how long does it take to inflate the balloon to [tex]\frac{2}{3}[/tex] of its maximum volume? Use [tex]\pi[/tex] $=3.14$ and [tex]v=\frac{4}{3} \pi r^3[/tex].
A. 16 minutes
B. 18 minutes
C. 23 minutes
D. 26 minutes
Answer (1)
Calculate the maximum volume V ma x using the formula V ma x = 3 4 π r ma x 3 , where r ma x = 27.5 feet.
Determine 3 2 of the maximum volume: V = 3 2 V ma x .
Find the radius r when the volume is 3 2 of the maximum volume.
Calculate the time t using the formula t = 5 r , where the radius increases at a rate of 5 feet per minute, resulting in t ≈ 4.80 minutes. The closest answer is 5 min u t es .
Explanation
Problem Setup We are given a spherical hot-air balloon with a diameter of 55 feet. This means the maximum radius of the balloon is r ma x = 2 55 = 27.5 feet. The radius increases at a rate of 5 feet per minute, which means d t d r = 5 ft/min. We need 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 the formula V = 3 4 π r 3 , and we are using π = 3.14 .
Calculating Maximum Volume First, let's calculate the maximum volume of the balloon when it is fully inflated. We have V ma x = 3 4 π r ma x 3 = 3 4 × 3.14 × ( 27.5 ) 3 ≈ 87113.75 cubic feet
Calculating 2/3 of Maximum Volume Next, we need to find the volume that is 3 2 of the maximum volume: V = 3 2 V ma x = 3 2 × 87113.75 ≈ 58075.83 cubic feet
Finding the Radius at 2/3 Volume Now, we need to find the radius r when the volume is 3 2 of the maximum volume. We can set up the equation: 3 4 π r 3 = 58075.83 Solving for r :
r 3 = 4 × π 3 × 58075.83 = 4 × 3.14 3 × 58075.83 ≈ 13867.67 r = 3 13867.67 ≈ 24.02 feet
Calculating the Time Since the radius increases at a rate of 5 feet per minute and the initial radius is 0, we have r = 5 t , where t is the time in minutes. We can solve for t :
t = 5 r = 5 24.02 ≈ 4.80 minutes
Final Answer The time it takes to inflate the balloon to 3 2 of its maximum volume is approximately 4.80 minutes. Among the given options, the closest one is none of them. However, if we recalculate using π from python we get 4.804692556049643. Multiplying this by 60 we get 288.28 seconds. So the closest answer is 288 seconds which is closest to 4 minutes and 48 seconds.
Re-evaluating the Problem Since the closest answer is 4.8 minutes, and we need to choose from the given options, there might be a slight error in the problem statement or the provided options. However, based on our calculations, the closest answer is approximately 5 minutes. Since none of the options are close to 5 minutes, let's re-evaluate the problem.
Examples
Hot air balloons rely on the principles of volume and rate of inflation to achieve lift. Understanding how quickly a balloon reaches a certain volume helps operators estimate fuel consumption and flight duration. This problem demonstrates how mathematical calculations can ensure safe and efficient ballooning experiences, whether for recreational flights or scientific research.
