The diagram shows a crane which has lifted a load from the ground. Loads are lifted and lowered vert

Question

The diagram shows a crane which has lifted a load from the ground. Loads are lifted and lowered vertically. The length of the arm of the crane is $a\text{ metres}$ and the angle the arm makes with the horizontal is $\theta^\circ$. The distance from the base of the cabin to the load when resting on the ground is $x\text{ metres}$.

Given that the height of the cabin is $2.5\text{ m}$, $x = 15\text{ m}$ and $a = 25\text{ m}$, find the height of the highest point of the crane from the ground.
[2]

The Safety Manual for the crane includes the following information:
The maximum load the crane can lift is $9000\text{ kg}$.
The table below shows the percentage of the maximum load that can be lifted when the length and angle of the arm are changed.

Arm length, $a$ (m)	Angle, $\theta$
	$20^\circ$	$40^\circ$	$60^\circ$	$80^\circ$
$15$	$65\%$	$80\%$	$90\%$	$100\%$
$20$	$50\%$	$65\%$	$80\%$	$90\%$
$25$	$30\%$	$50\%$	$65\%$	$80\%$
$30$	$10\%$	$30\%$	$50\%$	$65\%$

Use the information from the Safety Manual to help you answer the following questions.

Find the values of $a$ and $\theta$ for the crane to lift a load of $9000\text{ kg}$.
[1]
A load of $5850\text{ kg}$ is transported by the crane, and the crane is set to reach the maximum height allowed. Given that workers are not allowed to stand directly under the arm of a crane, work out the minimum distance that workers can safely stand from the cabin.
[4]

Accepted Answer

(a) $22.5\text{ m}$. (b) $a=15\text{ m}$, $\theta=80^\circ$. (c) Boundary distance $x=30\cos80^\circ\approx5.21\text{ m}$; stand beyond the arm’s horizontal reach.

Arm length, \(a\) (m)	Angle, \(\theta\)
	\(20^\circ\)	\(40^\circ\)	\(60^\circ\)	\(80^\circ\)
\(15\)	\(65\%\)	\(80\%\)	\(90\%\)	\(100\%\)
\(20\)	\(50\%\)	\(65\%\)	\(80\%\)	\(90\%\)
\(25\)	\(30\%\)	\(50\%\)	\(65\%\)	\(80\%\)
\(30\)	\(10\%\)	\(30\%\)	\(50\%\)	\(65\%\)

Trigonometry - Applications of Trigonometric Ratios

Trigonometry - Applications of Trigonometric Ratios

Solution:

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