Question

mposition of functions and modeling 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\\). 18 minutes 23 minutes 16 minutes 26 minutes

Explanation:

Step1: Find max radius

The diameter is 55 ft, so maximum radius $r_{max} = \frac{55}{2} = 27.5$ feet.

Step2: Calculate max volume

Use sphere volume formula $V_{max} = \frac{4}{3}\pi r_{max}^3$
$V_{max} = \frac{4}{3} \times 3.14 \times (27.5)^3$
$V_{max} = \frac{4}{3} \times 3.14 \times 20796.875 \approx 87173.83$ cubic feet

Step3: Find target volume

Target volume is $\frac{2}{3}$ of max volume:
$V_{target} = \frac{2}{3} \times V_{max} = \frac{2}{3} \times 87173.83 \approx 58115.89$ cubic feet

Step4: Solve for target radius

Rearrange volume formula to solve for $r_{target}$:
$r_{target} = \sqrt[3]{\frac{3V_{target}}{4\pi}}$
$r_{target} = \sqrt[3]{\frac{3 \times 58115.89}{4 \times 3.14}} = \sqrt[3]{\frac{174347.67}{12.56}} \approx \sqrt[3]{13881.2}$
$r_{target} \approx 24.0$ feet

Step5: Calculate radius increase needed

Initial radius is 0 (uninflated), so total radius increase:
$\Delta r = 24.0 - 0 = 24.0$ feet

Step6: Calculate inflation time

Time = $\frac{\text{Total radius increase}}{\text{Rate of increase}} = \frac{24.0}{1.5} = 16$ minutes

Answer:

16 minutes