What is the distance Y Z in metres between the boat and the bouy, to 2 significant figures?

A surveyor, standing at an observation point on a cliff top of a height 300m above sea level, can see a boat and a bouy out at sea, both directly in front of him. He measures the angle from the top of the cliff to the bouy Y as 27 degrees and the angle from the top of the cliff to the boat Z as 18 degrees

make a diagram and put in all your information.
You should see two right-angled triangles.

tan 27 = 300/x, where x is the distance from the base of the cliff to the bouy is your first calculation.

if y is the distance from the base of the cliff to the boat, then tan 18 = 300/y

evaluate x and y, take y-x and state the answer to 2 significant figures.
Let me know what you get.

Reiny thank you so much for helping me.
Is it 55 ??

no, from tan 27 = 300/x
I got x= 300/tan27 = 588.8

from tan 18 = 300/y I got
y=300/tan 18 = 923.3

then the difference is 923.3-588.8
=334.5
which is 330 m to 2 significant figures.

You're welcome! I'm glad I could help. However, the answer you provided, 55 meters, is incorrect. Let's go through the calculations together to find the correct answer.

To solve for the distance from the base of the cliff to the buoy (denoted as x), we can use the tangent function. The given information is that the angle from the top of the cliff to the buoy is 27 degrees, and the height of the cliff is 300 meters.

We can set up the equation as follows: tan(27) = 300/x

To solve for x, we need to isolate it on one side of the equation. Taking the inverse tangent (arctan) of both sides will help us do that. Remember to use a calculator in degree mode:

x = 300/tan(27)
x ≈ 588.8 meters (rounded to one decimal place)

Next, let's solve for the distance from the base of the cliff to the boat (denoted as y). The given information is that the angle from the top of the cliff to the boat is 18 degrees, and the height of the cliff remains 300 meters.

Setting up the equation: tan(18) = 300/y

Again, isolate y by taking the inverse tangent of both sides:

y = 300/tan(18)
y ≈ 923.3 meters (rounded to one decimal place)

Finally, to find the distance between the boat and the buoy (denoted as YZ), subtract the distance from the base of the cliff to the buoy (x) from the distance from the base of the cliff to the boat (y):

YZ = y - x
YZ ≈ 923.3 - 588.8
YZ ≈ 334.5 meters (rounded to one decimal place)

Therefore, the distance YZ between the boat and the buoy, to two significant figures, is approximately 330 meters.