Question

Matt is standing on top of a cliff 305 feet above a lake. The measurement of the angle of depression to a boat on the lake is 42. How far is the boat from Matt

Answer 1

Answer:

≈ 410.4427 feet (this is rounded -- it's an infinite decimal)

Step-by-step explanation:

If you draw it out, you get a right triangle. If Matt is standing 305 feet above a lake, that is the altitude of the triangle. The angle of depression means, for example, the angle looking down from something at something else, in this case, from the cliff to the boat. The 42° is the acute angle between the altitude and the hypotenuse, the one at the top of the triangle. You want to find how far away the boat is from Matt directly, so that would be the hypotenuse, just connecting a straight line from him to the boat.

To answer these problems there are three tools you use to find the answer: Sine, Cosine, and Tangent. Let's see which one would best help us in this situation.

The definition of sine is the side opposite the acute angle over the hypotenuse.

The definition of cosine is the side adjacent to the acute angle over the hypotenuse.

The definition of tangent is the side opposite the acute angle over the side adjacent to it.

In this case, you know you have the side adjacent to the acute angle, over the hypotenuse. Because you have the acute angle (42°) and the side adjacent to it (305) and the hypotenuse (which is what you're trying to find the value of. Which definition would fit this figure? Cosine, because it's side adjacent over hypotenuse.

Once you have that straightened out, you know that you need to find the cosine of 42°, which, if you look in a sine, cosine, and tangent chart (or type 42 into your calculator and hit cos) is 0.7431. Now you can write the equation $0.7431 = \frac{305}{x}$.

Working with that equation, you need to get x by itself to figure out its value. How you do that is by multiplying both sides by x. This would look like $0.7431 * x = 305 * x$. When you multiply 305 by x, you are multiplying by the number you were just dividing by so that answer is 305. I'll give you an example with different numbers: 6/2 then 6*2 is just 6.

What your equation looks like now is $0.7431 * x = 305$. How do you find x? Well it's the same as doing this: $2 * x = 8$. Then you would divide 8 by 2 to get x, which equals 4. So, going back to our task at hand, $305/0.7431 = x = 410.4427 (approximately)$. That's your answer, which is how far Matt is from the boat. Depending on which digit they want you to round the answer to you could punch in 305/0.7431 into a calculator and see many more digits and a much longer number. But don't forget to include units in your answer -- feet.