Hot air balloon

A hot air balloon ascends 25 meters for a minute after launch. Every minute, it ascends 75 percent of the height it climbed in the previous minute.

a) How many meters will ascend six minutes after takeoff?
b) What is the overall height of the 10 minutes after launch?
c) how long does the balloon take to gain over 110 meters high?
d) What will the final height of the balloon be?

Correct answer:

a =  5.93 m
b =  94.37 m
t =  NAN min
d =  100 m

Step-by-step explanation:

$$\begin{gathered} q=75/100=0.75 \\ \Delta s(t)=25\cdot 0.75^{t-1} \\ a=\Delta s(6)=25\cdot 0.75^{6-1}=5.93\text{m} \end{gathered}$$

$b=s_6=a_1\frac{q^6-1}{q-1}=25\cdot \frac{0.75^6-1}{0.75-1}=94.37\text{m}$

$$\begin{gathered} s_t=25\frac{q^t-1}{q-1}=110 \\ {q}^t-1=110(q-1)/25 \\ {q}^t=110(q-1)/25+1 \\ t\log q=\log (110\cdot (q-1)/25+1) \\ t=NAN\text{min} \end{gathered}$$

$d=s_{\inf }=\frac{a_1}{1-q}=\frac{25}{1-0.75}=100\text{m}$

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