(a) If \(\log_{10} 2 = 0.3010\) and \(\log_{10} 3 = 0.4771\), calculate without using tables, the value of \(\log_{10} 0.72\). (b) A hawk on top of a tree, ...
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(a) If \(\log_{10} 2 = 0.3010\) and \(\log_{10} 3 = 0.4771\), calculate without using tables, the value of \(\log_{10} 0.72\).
(b) A hawk on top of a tree, 20 metres high views a chick on the ground at an angle of depression of 39°. Find, correct to 2 significant figures, the distance of the chick from the bottom of the tree.
Answer Details
(a) Express 0.72 using 2 and 3: \(0.72 = \dfrac{72}{100} = \dfrac{2^3 \times 3^2}{10^2}\).
\[ \log_{10} 0.72 = 3\log_{10}2 + 2\log_{10}3 - 2\log_{10}10. \] \[ = 3(0.3010) + 2(0.4771) - 2(1) = 0.9030 + 0.9542 - 2 = 1.8572 - 2. \] \[ \log_{10} 0.72 = -0.1428. \](b) The tree is 20 m high; the chick is on the ground at angle of depression \(39^\circ\) from the top. The angle of depression equals the angle of elevation from the chick, so in the right-angled triangle (height 20 m, horizontal distance \(d\)):
\[ \tan 39^\circ = \frac{20}{d} \implies d = \frac{20}{\tan 39^\circ} = \frac{20}{0.8098} = 24.7\ \text{m}. \]The distance of the chick from the bottom of the tree is 25 m (to 2 significant figures).
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