A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly. When first noticed the angle of depression to the boat is 16°18'. When the boat stops, the angle of depression is 48°51'. The lighthouse is 200 feet tall. How far did the boat travel from when it was first noticed until it stopped? Round your answer to the hundredths place.

I kinda got a hold of certain aspects of the questions but I can't get the math part right :( I'll give medal ofc.

Question

A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly. When first noticed the angle of depression to the boat is 16°18'. When the boat stops, the angle of depression is 48°51'. The lighthouse is 200 feet tall. How far did the boat travel from when it was first noticed until it stopped? Round your answer to the hundredths place.

I kinda got a hold of certain aspects of the questions but I can't get the math part right :( I'll give medal ofc.

anonymous · Answer

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Might this help?

anonymous · Answer

That helps! 16 18' turns into 16.3 and 48 51' turns into 48.85, right? And then I use tan?

jdoe0001 · Answer

yes

anonymous · Answer

Yeah, you have the minutes to decimal conversion correct, and you use tan because you don't care about they hypotenuses. Remember you know the length of the opposite side to the known angles, rather than the adjacent.

jdoe0001 · Answer

tangent  of one angle, get the adjacent side
tangent of the other angle, get the adjacent side

the difference between both adjacent sides found, is how long the boat went

anonymous · Answer

Okay, good. I think it's something like 200/tan(16.3) and 200/(48.85) but once I put it into my calculator I get some weird large numbers and I get lost after that.

anonymous · Answer

You shouldn't be getting any large numbers
$$\frac{200}{\tan(16.3^\circ)} = 683.9$$
$$\frac{200}{\tan(48.85^\circ)}= 174.78$$

anonymous · Answer

Are you suppose to divide 200 by tan(16.3) and etc?

anonymous · Answer

tan(angle) is opposite over adjacent and you know the opposite, so dividing opposite by tan(angle) will give the adjacent

anonymous · Answer

I put in 200/tan(16.3) into my calculator and got 297.3935743???

anonymous · Answer

Your calculator is set to radians

anonymous · Answer

Can you fix that or something?

anonymous · Answer

Most calculators will let you change to degrees. If it's a scientific calculator there should be a menu button somewhere. If you can't, the conversion is
$$degrees = \frac{360}{2\pi} \times radians$$

anonymous · Answer

I fixed it thanks! After that would I be subtracting 683.9 from 174.78?

anonymous · Answer

Yep

anonymous · Answer

So your answer ends up being 509.12 and that's how far it traveled?

anonymous · Answer

Yeah, unless we've made a silly mistake somewhere.

The distances look sensible, so I'm sure it's right.

anonymous · Answer

Gotcha. Thank you so much!!!