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Completion requirements
The arm of a picker truck has a maximum extension of \(6\) m.
Determine the exact horizontal distance the bucket of the picker truck
moves as it moves from an angle of elevation of \(30^\circ\) to
\(45^\circ\).
A diagram:
Determine the horizontal distance moved by the bucket.
\(\begin{align}
h &= h_1 - h_2 \\
h &= 3\sqrt 3 - 3\sqrt 2 \\
\end{align}\)
The horizontal distance moved by the bucket is \(3\sqrt{3} - 3\sqrt{2} \rm \thinspace m\).
\(\begin{align}
\cos \theta &= \frac{{{\mathop{\rm adj}\nolimits} }}{{{\mathop{\rm hyp}\nolimits} }} \\
\cos 30^\circ &= \frac{{h_1 }}{6} \\
\frac{{\sqrt 3 }}{2} &= \frac{{h_1 }}{6} \\
3\sqrt 3 &= h_1 \\
\end{align}\)
\cos \theta &= \frac{{{\mathop{\rm adj}\nolimits} }}{{{\mathop{\rm hyp}\nolimits} }} \\
\cos 30^\circ &= \frac{{h_1 }}{6} \\
\frac{{\sqrt 3 }}{2} &= \frac{{h_1 }}{6} \\
3\sqrt 3 &= h_1 \\
\end{align}\)
\(\begin{align}
\cos \theta &= \frac{{{\mathop{\rm adj}\nolimits} }}{{{\mathop{\rm hyp}\nolimits} }} \\
\cos 45^\circ &= \frac{{h_2 }}{6} \\
\frac{{\sqrt 2 }}{2} &= \frac{{h_2 }}{6} \\
3\sqrt 2 &= h_2 \\
\end{align}\)
\cos \theta &= \frac{{{\mathop{\rm adj}\nolimits} }}{{{\mathop{\rm hyp}\nolimits} }} \\
\cos 45^\circ &= \frac{{h_2 }}{6} \\
\frac{{\sqrt 2 }}{2} &= \frac{{h_2 }}{6} \\
3\sqrt 2 &= h_2 \\
\end{align}\)
\(\begin{align}
h &= h_1 - h_2 \\
h &= 3\sqrt 3 - 3\sqrt 2 \\
\end{align}\)
The horizontal distance moved by the bucket is \(3\sqrt{3} - 3\sqrt{2} \rm \thinspace m\).