[SOLVED] Sine Question

Mar 2010
On a sunny day, a tower casts a shadow 35.2m long. At the same time, a 1.3m parking meter that is nearby casts a shadow 1.8m long. How high is the tower to the nearest tenth of a metre?

Triangle formed by parking meter:
\(\displaystyle c^2 = a^2 + b^2\)
\(\displaystyle c^2 = 1.3^2 + 1.8^2\)
\(\displaystyle c^2 = \frac{493}{100}\)
\(\displaystyle c = \frac{\sqrt{493}}{10}\)

\(\displaystyle SinA = \frac{O}{H}\)
\(\displaystyle SinA = \frac{1.3}{\frac{\sqrt{493}}{10}}\)
\(\displaystyle Sin^{-1}\frac{1.3}{\frac{\sqrt{493}}{10}} = A\)
\(\displaystyle 35.84^{\circ} = A\)

Triangle formed by tower (Total triangle):
\(\displaystyle \frac{a}{SinA} = \frac{b}{SinB}\)
\(\displaystyle \frac{a}{Sin35.84} = \frac{37}{Sin54.16}\)
\(\displaystyle a = Sin35.84(\frac{37}{Sin54.16})\)
\(\displaystyle a = 26.7m\)

*The \(\displaystyle 37m\) value was obtained by adding \(\displaystyle 35.2m\) and \(\displaystyle 1.8m\).
*The \(\displaystyle 54.16^{\circ}\) value was obtained by subtracting \(\displaystyle 90deg\) and \(\displaystyle 35.84deg\) from \(\displaystyle 180deg\).

Therefore, the height of the tower is \(\displaystyle 26.7m\).

Textbook answer: \(\displaystyle 25.4m\).

Where did I make the mistake?


MHF Helper
Jan 2008
Big Stone Gap, Virginia
Hi RogueDemon,

Actually, you made this way harder than it is.

You have two similar right triangles.

Set up a proportion using the ratios of the heights and shadows.

\(\displaystyle \frac{bldg \:\: ht.}{1.3}=\frac{35.2}{1.8}\)
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