^{2}= A

^{2}+ B

^{2}-2*A*B*cos φ; φ = 70 degrees, A=200, B=452. plugging into the equation we obtain:

Sarah is walking around the lake and stops to take measurements of the water fountain. She is at a bearing of 230 degrees from the fountain. The angle of elevation at this point is 23 degrees from a distance of 200m away. She again takes a measurement from the other side of the lake. At this point, her bearing to the fountain is 300degrees and the angle of elevation to the fountain is 11 degrees. She is 452m away. Calculate, how far Sarah is from her original position and the height of the fountain on that day? (Assume she is on the same level at both calculations.) draw a diagram.

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The angle between the two visual points is 300-230 = 70 degrees

the two sides of the triangle that conforms the two visuals are 200m and 452 m, so applying the cosine theorem we have C^{2} = A^{2} + B^{2} -2*A*B*cos φ; φ = 70 degrees, A=200, B=452. plugging into the equation we obtain:

C= 427.16 m

Now, the altitude of the fountain is Tan23=h/200 => h=200*tan 23 =84.89 m first day

to check I use that tan 11= h/452 => h=452*tan 11 = 87.85m. second day

Lilly,

Unfortunately we can't share diagrams for solving this problem in this format, but the best way for most people to solve it is by drawing it out. So let me try to explain visually how it would look.

For the height of the fountain, draw a right triangle with hypotenuse sloping up to the right and the longer of the two sides of the 90 deg angle on the bottom. The angle of the slope should be about 23 deg in your diagram. The hypotenuse in this triangle represents the line of sight, the longer of the two sides of the 90 deg angle on the bottom is the distance to the fountain, and the side opposite of the 23 deg angle is the height of the fountain. According to this diagram, the height of the fountain (side opposite) divided by the distance to the fountain (side adjacent) is equal to the tangent of the angle. Putting what I just said into equation form, tan(23 deg) =0.42447 = x/200 where x = height of fountain. Rearranging, x = 200(0.42447) = 84.89 meters. You should try to repeat this example using data for the second point.

To determine how far the two observation points are from each other, you can use the formula recommended by Francisco in the first answer. Or you can diagram it by mapping out both points on an xy coordinate graph and either measuring or calculating the distance between them. So let's go.

The bearing from the fountain to the first point is 230 deg, meaning that the point is in the southwest quadrant from the fountain when the fountain is at 0,0. Mark that on your diagram. Draw a line from the observation point to the fountain to represent the 200 m distance to the fountain. Next you need to get the x and y components of the distance. If the distance to the fountain was 1.0, then the x component would be the cosine of the angle which is 230 deg (recall the unit circle). But since the distance is 200 m, the x component is 200(cos 230) = 200(-0.6428) = - 128.6 m. The y component will be 200(sin 230) = 200(-0.7660) = -153.2 m. What this means is that with the fountain at the origin, the coordinates of the first observation point in meters will be -128.6, -153.2.

From the second observation point, the bearing to the fountain is 300 deg, meaning the point is in the southeast quadrant when the fountain is at the origin. Mark a point in the southeast quadrant which is about twice as far from 0,0 as the point you just did in the southwest quadrant and connect it to 0,0 with a line representing the distance to the fountain = 452 m. Now the x component is 452(cos 300 deg) = 452(0.5) = 226.0 m, and the y component is 452(sin 300 deg) = 452(-0.8660) = -391.4 m. The coordinates of the second point are 226.0, -391.4.

Now all you need to do is find the distance between the two points using the Pythagorean theorem. The x distance from point 1 to point 2 is 226.0 - (-128.6) = 354.6 m and the y distance between the two points is -391.4 - (-153.2) = -238.2 m. Then the distance = ((354.6)^2 + (-238.2)^2)^0.5 = (182,480)^0.5 = 427.2 m.

Hope you enjoy this alternate explanation in case you are still interested in this problem.

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