a person measures the angle of depression from the top of a wall to a point on the ground. The point is located on level ground 62 feet from the base of the wall and the angle of depression is 52 degrees. How high is the wall, to the nearest tenth of a foot? Include a drawing of your work.

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Not sure what you mean by angle of depression, so I am imagining that there's a triangle made by connecting the top of the wall, the bottom of the wall, and the point located on level ground 62 feet from the base of the wall.  

Assume the wall is plumb (i.e. 90 degree angle between the wall and the level ground).  If the "angle of depression" is measured from the top of the wall to the distant point at 52 degrees, that means the angle between the distant point and the top of the wall is:

180 - ( 90 + 52 ) = 180 - 142 = 38 degrees

because in every triangle, the angles add up to 180 degrees.

Now, if I have misunderstood "angle of depression" and placed it in the wrong part of the triangle, the angles are still 38, 52 and 90 degrees, just swap the position of the 38 and 52 degree angles.  

So now we have all three angles and one side (the 62 feet between the bottom of the wall and the distant point - consider this the side "adjacent" to the distant point). It is now possible to find the two remaining sides using trigonometric relationships.  

Let's continue the assumption that the angle from the distant point to the top of the wall is 38 degrees, call that theta.  That means that the hypotenuse of the right triangle is the long side of the triangle from the distant point to the top of the wall. In an right triangle, the cosine of the angle is the ratio of the adjacent side to the hypotenuse. So if we are finding the hypoteneuse, it is given by the formula

hypotenuse = adjacent / cos ( theta )

Where cos is the cosine function, evaluated in degrees. cos(38) = 0.788, so

hypotenuse = 62 feet / 0.788 = 78.769 feet

The height of the wall - the side opposite to theta - is found from the fact that the sine of the angle theta is the ratio of the opposite side to the hypotenuse. So we can find the opposite side by  

opposite = hypotenuse * sin ( theta )  

sin (38) = 0.615, so  

opposite = 78.679 feet * 0.615 = 48.44 feet.

If I misinterpreted "angle of depression," follow the same logic only use theta=52. The height of that wall would be 79.35 feet.

Hope this helped you with you question!

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the figures below are made out of circles, semicircles, quarter … find the area and the perimeter of each figure and give your answers as a completely simplified exact value in terms of π (no approximations). remember that we'll also have to cut whatever areas we get in half, since we're only dealing im pretty sure that is the right answer

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