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Answer :
Let P and Q be the two positions of the balloon and let A be the point of observation. let ABC be the horizontal through A. It is given that angles of elevation of the balloon in two position P and Q from \( ∠ \ PAB \ = \ 60º \) , \( ∠ \ QAB \ = \ 30º \). It is also given that MQ = 88.2
\( => \ CQ \ = \ MQ \ - \ MC \ \)
\( = \ 88.2 \ - \ 1.2 \ = \ 87 \)m .
In \( ∆ \ ABP \), we have
\( \frac{BP}{AB} \ = \ tan60º \ \)
\( => \ \frac{87}{AB} \ = \ \sqrt{3} \)
\( => \ AB \ = \ \frac{87}{ \sqrt{3}} \ = \ \frac{87 \sqrt{3}}{3} \ = \ 29 \sqrt{3} \) (1)
In \( ∆ \ ACQ \), we have
\( \frac{CQ}{AC} \ = \ tan30º \ \)
\( => \ \frac{87}{AC} \ = \ \frac{1}{ \sqrt{3}} \)
\( => \ AC \ = \ 87 \sqrt{3} \) (2)
\(\therefore \) \( PQ \ = \ BC \ = \ AC \ - \ AB \ \)
\( = \ 87 \sqrt{3} \ - \ 29 \sqrt{3} \ \)
\( = \ 58 \sqrt{3} \)
\( = \ 100.456 \)m
\(\therefore \) the balloon travels \( 100.456 \)m