# Hot air balloon

Hot air balloon ascends 25 meters up for a minute after launch. Every minute ascends 75 percent of the height which climbed in the previous minute.

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

Result

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

#### Solution:

$q = 75/100 = 0.75 \ \\ \Delta s(t) = 25\cdot 0.75^{ t- 1} \ \\ a = \Delta s(6) = 25\cdot 0.75^{ 6- 1} = 5.93 \ \text{m}$
$b = s_6 = a_1 \dfrac{q^6-1}{q-1} = 25 \cdot \dfrac{ 0.75^6-1}{ 0.75-1} = 94.37 \ \text{m}$
$s_t = 25 \dfrac{q^t-1}{q-1} = 110 \ \\ q^t-1 = 110(q-1)/ 25 \ \\ q^t = 110(q-1)/ 25 + 1 \ \\ t \log q = \log (110 \cdot (q-1)/ 25 + 1) \ \\ t = NAN \ \text{min} \ \\$
$d = s_{\inf} = \dfrac{ a_1 }{ 1-q} = \dfrac{ 25 }{ 1-0.75} = 100 \ \text{m}$

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