Example 3
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| Example 3 |
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When an earthquake occurs, different kinds of seismic waves are produced. Two types of seismic waves are S-waves and P-waves, which travel at different velocities. A seismic station can estimate the distance to an earthquake’s epicentre by comparing the amount of time that elapses between the two types of waves reaching the station.
An earthquake’s epicentre occurs \(48 \thinspace \rm{km}\) away from seismic station 1, and \(91 \thinspace \rm{km}\) away from seismic station 2. If the seismic stations are known to be \(112 \thinspace \rm{km}\) apart, what is the angle formed at seismic station 2 between seismic station 1 and epicentre 1?
An earthquake’s epicentre occurs \(48 \thinspace \rm{km}\) away from seismic station 1, and \(91 \thinspace \rm{km}\) away from seismic station 2. If the seismic stations are known to be \(112 \thinspace \rm{km}\) apart, what is the angle formed at seismic station 2 between seismic station 1 and epicentre 1?
There are three known sides and an unknown angle involved, so you can use the cosine law. It helps to draw a diagram.
Write down what is given.
\(\angle C = \thinspace ?\)
\(a = 112\thinspace \rm{km}\)
\(b = 91\thinspace \rm{km}\)
\(c = 48\thinspace \rm{km}\)
\(\begin{align}
\cos C &= \frac{{a^2 + b^2 - c^2 }}{{2ab}} \\
\cos C &= \frac{{\left( {112} \right)^2 + \left( {91} \right)^2 - \left( {48} \right)^2 }}{{2\left( {112} \right)\left( {91} \right)}} \\
\cos C &= 0.908... \\
\angle C &= \cos ^{ - 1} \left( {0.908...} \right) \\
\angle C &= 24.686...^\circ \\
\angle C &\doteq 24.7^\circ \\
\end{align}\)
The angle formed at seismic station 2, between seismic station 1 and epicentre 1, is approximately \(24.7^\circ\).
Write down what is given.
\(\angle C = \thinspace ?\)
\(a = 112\thinspace \rm{km}\)
\(b = 91\thinspace \rm{km}\)
\(c = 48\thinspace \rm{km}\)
\(\begin{align}
\cos C &= \frac{{a^2 + b^2 - c^2 }}{{2ab}} \\
\cos C &= \frac{{\left( {112} \right)^2 + \left( {91} \right)^2 - \left( {48} \right)^2 }}{{2\left( {112} \right)\left( {91} \right)}} \\
\cos C &= 0.908... \\
\angle C &= \cos ^{ - 1} \left( {0.908...} \right) \\
\angle C &= 24.686...^\circ \\
\angle C &\doteq 24.7^\circ \\
\end{align}\)
The angle formed at seismic station 2, between seismic station 1 and epicentre 1, is approximately \(24.7^\circ\).