Question

a hot air balloon is at an altitude of 100 1\5 yards, the balloons altitude decreases by 10 4/5 every minute. determine the number of minutes it will take the balloon to reach an altitude of 57 yards.

Answer 1

To determine the number of minutes it will take for the hot air balloon to descend from an altitude of \(100 \frac{1}{5}\) yards to \(57\) yards while descending at a rate of \(10 \frac{4}{5}\) yards per minute, we can follow these steps:

1. **Convert the mixed numbers to improper fractions.**

- For \(100 \frac{1}{5}\):
\[
100 \frac{1}{5} = 100 + \frac{1}{5} = \frac{100 \times 5 + 1}{5} = \frac{500 + 1}{5} = \frac{501}{5}
\]

- For \(10 \frac{4}{5}\):
\[
10 \frac{4}{5} = 10 + \frac{4}{5} = \frac{10 \times 5 + 4}{5} = \frac{50 + 4}{5} = \frac{54}{5}
\]

2. **Calculate the altitude difference:**
\[
\text{Altitude difference} = 100 \frac{1}{5} - 57 = \frac{501}{5} - 57
\]
Convert \(57\) to a fraction:
\[
57 = \frac{57 \times 5}{5} = \frac{285}{5}
\]

Now calculate:
\[
\frac{501}{5} - \frac{285}{5} = \frac{501 - 285}{5} = \frac{216}{5}
\]

3. **Calculate the number of minutes to descend this altitude at the given rate:**
We set up the equation:
\[
\text{Time} = \frac{\text{Altitude difference}}{\text{Rate of descent}} = \frac{\frac{216}{5}}{\frac{54}{5}}
\]
When dividing fractions, we multiply by the reciprocal:
\[
\text{Time} = \frac{216}{5} \times \frac{5}{54} = \frac{216}{54} = 4
\]

Thus, it will take **4 minutes** for the hot air balloon to reach an altitude of 57 yards.