When does the hot air balloon reach a height of 5000m?

Question

In a math problem using geometric series they ask the following. When a hot air balloon for weather measurements is launched he rises 500m in the first minute. Every following minute the balloon rises 12% less then the minute before.

First they asked the height of the balloon after 10 minutes, so because $t_1 = 500$ and $q=0,88$ I found $t_{10} = 3006,2459...m$, so roughly 3006m.

Next they ask us when the balloon will reach a height of 5000m. Using the same elements I found in the previous part I find that $0,2 = -0,88^n$. Wich makes me think there is no real solution to this question.

I can't believe our lector would ask us a questoin without an answer so I want to quickly check if I made a mistake somewhere.

Answer 1

The balloon rises $r_k := 500 \cdot (1-0.12)^k$ between minutes $k$ and $k+1$. Assuming that the balloon starts at zero altitude, the maximum altitude reached is

$$\sum_{k=0}^{\infty} r_k = 500 \sum_{k=0}^{\infty} (1-0.12)^k = \frac{500}{0.12} \approx 4166.67 < 5000$$

Answer 2

You are right the limiting value ie highest value attained is $s= 500/(0.12) \approx 4166m$

Answer 3

To see that it never rises to this level one simply needs to compute the following infinite geometric series.

$500\sum_{n=0}^{\infty}0.88^n = \frac{500}{1-0.88}=4166.\bar{6}$

Let me know if you need any further details!