Hi! The question is really confusing to me. I can't sketch/visualize the problem. :(

From a given position an observer notes that the angle of elevation of a rock is 47 degrees. After walking 1000 feet towards the rock, up a slope of 32 degrees, he finds the angle of elevation to be 75 degrees. Find the vertical distance of the rock above each point of observation.

Label the two observation points A,B and the rock R.

Drop a vertical from R and draw horizontals from A,B to intersect it at M,N. So, AM and BN are horizontal lines.
Let RN intersect the slope at P.

Drop a vertical from B to intersect AM at C.

We want the distances RN and RM

RN/BN = tan 75°
RM/AM = tan 32°
AM = BN + 1000 cos47°
RM = RN + 1000 sin47°

That should get you started. Let us know how things go.

Oh, I forgot, we were supposed to use the Sine Law. Sorry :3

Well, don't worry! I'll try to paint a funny picture for you. Imagine you're playing hide-and-seek with a rock. The rock is hiding on top of a hill, and you're trying to find it.

First, you start at a certain position and look up at the rock. The angle of elevation is 47 degrees. But the rock is a sneaky one and wants to make it challenging for you. So, you decide to walk 1000 feet closer to it. You're determined!

As you make your way up the slope, which is at a delightful angle of 32 degrees, you keep your eyes on the rock. Finally, you reach a spot where you can see the rock clearly again. Now the angle of elevation has increased to 75 degrees. Success!

But now, the big question is: How high did the rock jump while you were approaching it? We need to find the vertical distance of the rock above each observation point.

To solve this, let's call the vertical distance of the rock above the first observation point "x." And let's call the vertical distance of the rock above the second observation point "y." Now, we have two equations to work with. Are you ready?

The first observation angle is 47 degrees, so we can say that:

x / 1000 = tan(47 degrees)

The second observation angle is 75 degrees, and because you've walked 1000 feet closer, we can say that:

y / 0 = tan(75 degrees - 32 degrees)

(Note: The second equation has 0 because this is the reference point where you're standing.)

Don't worry, I won't leave you hanging here. We can simplify those equations and find the values of "x" and "y" that will reveal the vertical distance of the rock above each observation point. Keep rocking, and remember, hills and rocks can always use a little more humor!

No worries, I can help you visualize and solve the problem step by step!

To make it easier to understand, let's break down the problem into two parts:

1. Finding the vertical distance between the initial observation point and the rock.
2. Finding the vertical distance between the point after walking towards the rock and the rock itself.

Let's start with the first part:

1. Finding the Vertical Distance from the Initial Observation Point to the Rock:

When the observer is at the initial observation point, they measure the angle of elevation to be 47 degrees. Let's say this distance is 'x' feet.

To visualize this, imagine a right-angled triangle where:
- The vertical side represents the vertical distance from the observation point to the rock (which we need to find).
- The horizontal side represents the horizontal distance between the observation point and the rock (which is 1000 feet).
- The hypotenuse represents the straight-line distance between the observation point and the rock, which we don't need to consider right now.

Using trigonometry, we can use the tangent function to find the vertical distance:
tan(47 degrees) = vertical distance / 1000 feet

To find the vertical distance, we rearrange the equation:
vertical distance = tan(47 degrees) * 1000 feet

Now, if you have a scientific calculator or access to trigonometric tables, you can directly calculate the value of tan(47 degrees) and then multiply it by 1000. The product will give you the vertical distance from the initial observation point to the rock.

Moving on to the second part:

2. Finding the Vertical Distance from the Point after Walking towards the Rock to the Rock:

When the observer walks 1000 feet towards the rock, they measure the angle of elevation to be 75 degrees. Let's say this new vertical distance is 'y' feet.

Again, visualize a right-angled triangle, where the vertical side represents the vertical distance from the observer's new position to the rock (which we need to find). The horizontal side represents the horizontal distance (which is still 1000 feet), and the hypotenuse represents the straight-line distance (which we don't need to consider right now).

Using trigonometry, we can again use the tangent function to find the vertical distance:
tan(75 degrees) = vertical distance / 1000 feet

Rearranging the equation to solve for the vertical distance:
vertical distance = tan(75 degrees) * 1000 feet

Similar to before, you can calculate the value of tan(75 degrees) and then multiply it by 1000 to get the vertical distance from the point after walking towards the rock to the rock.

I hope this step-by-step explanation helps you understand how to approach and solve the problem!