[SOLVED] Sine Question

Mar 2010
78
2
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?
 

masters

MHF Helper
Jan 2008
2,550
1,187
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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