Help needed on trigonometry question

There is a fire in a 40-metre-tall apartment building. A group of people are stuck on the rooftop. Two firetrucks are speeding towards the building. From the top of the building Firetruck 1 is at an angle of depression of 15 degrees and the angle of depression to Firetruck 2 is 22 degrees. From the bottom of the building the angle between the two firetrucks is 32 degrees. How far apart are the two firetrucks?

I’m not sure how to draw a diagram for this question, and I’m also confused by the 32 degree angle in between the 2 trucks.

I also had a tough time interpreting the question, it’s kinda weird. But I think I’ve figured it out.

It turns out that there is not one diagram that will model this whole question, but rather you’ll have to draw out a few diagrams.

STEP 1: Find the distance from the building to firetruck 1 (call this distance x_1). And find the distance from the building to firetruck 2 (call this distance x_2).

So “angle of depression” means, from the people at the top of the building’s perspective, looking down 15^\circ at firetruck 1, and looking down 22^\circ at firetruck 2. So this means that the angle between firetruck 1 and the top of the building is (90^\circ - 15^\circ) = 75^\circ, and the angle between firetruck 2 and the top of the building is (90^\circ - 22^\circ) = 68^\circ.

Now, since we know that the height of the building is 40 meters, we can use the tangent function to find the horizontal distances x_1 and x_2:

\tan (75^\circ) = \dfrac{x_1}{40} \implies x_1 = 40\tan (75^\circ) \approx 149.28\tag*{}
\tan (68^\circ) = \dfrac{x_2}{40} \implies x_2 = 40\tan (68^\circ) \approx 99\tag*{}

STEP 2:
Do you know what the law of cosines is? This allows you to calculate the third side of a triangle given that you know what two sides are and the angle between them. In our case, we have calculated x_1 and x_2, and we know that the horizontal angle between them is 32^\circ. This gives us enough information to calculate the distance d between the two trucks, using the law of cosines:

\begin{align*} d &= \sqrt{(x_1)^2 + (x_2)^2 - 2(x_1)(x_2)\cos (32^\circ)} \\ &= \sqrt{(149.28)^2 + (99)^2 - 2(149.28)(99)\cos (32^\circ)} \\ &\approx 83.78 \end{align*}\tag*{}

Here are my diagrams and work:

Thank you so much. I appreciate all your help and am so thankful that you chose to help me. This actually makes so much more sense with multiple diagrams, I didn’t even think of that! Thanks again.

I’m very happy to help! If you know anyone else who might find use of my math help too, be sure to let them know! Spread the word : )

I will be sure to! Thanks again!

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