Question

The ramp for uploading a moving truck, has an angle of elevation of $30{}^\circ $ . If the top of the ramp is 0.9m above the ground level, then find the length of the ramp.

Answer 1

Hint: Assume the length of the ramp to be l meters and draw a rough diagram of the situation given in the question. Draw the diagram you will get a right angles triangle with ground as the base, ramp as the hypotenuse and the height equal to 0.9 m. So, as you know the height and one of the angles, use the formula $\sin \theta =\dfrac{height}{hypotenuse}$ to get the length of the ramp.

Complete step-by-step answer:
Let us start with the question by letting the length of the ramp be l meters and drawing a representative diagram of the situation given in the question.

According to the above figure:
BC is the ground and AC is the ramp..
Now, in right angle triangle ABC,
We have $\angle ACB=30{}^\circ $ .
We know that, $\sin \theta =\dfrac{height}{hypotenuse}$. And it is given that the height of the highest point of the ramp, i.e., the height is equal to 0.9m. So, we can say that:
$\begin{align}
  & \therefore \sin 30{}^\circ =\dfrac{AB}{AC} \\
 & \Rightarrow \sin 30{}^\circ =\dfrac{0.9}{l} \\
\end{align}$
Now we know that sine of $30{}^\circ $ is $\dfrac{1}{2}$ .
$\therefore \dfrac{1}{2}=\dfrac{0.9}{l}$
Now to get the value of l, we will cross-multiply. On doing so, we get
$l=0.9\times 2=1.8m$
Therefore, the length of the ramp is 1.8 m.

Note: For doing the question in the shortest possible manner you must be very careful in selecting the trigonometric ratio that you will use to solve the question. You can solve the above question by finding the base using a tangent of the known angle followed by Pythagoras theorem but that would be lengthier and complicated to solve as compared to the above solution. Remembering the values of the trigonometric ratios for standard angles is very important.