To estimate the height of a house Katie stood a certain distance from the house and determined that the angle of elevation to the top of the house was 32 degrees. Katie then moved 200 feet closer to the house along a level street and determined the angle of elevation was 42 degrees. What is the height of the house. Step by Step Explanation .. Please

Step-by-step explanation:

Let's call point A where Katie was standing when she started this whole mess. Let's call point B then where she ended up when she was 200 feet closer to the house.  y is the unknown height of the house, and x is the unknown number of feet she is standing from the house.

If we let x be the distance she starts off from the house (point A), when she moves closer (to point B), she is then taking 200 feet from the distance from point A.  In other words, from point A to point B is 200 feet (that's her moving 200 feet from A closer to the house, point B); and x - 200 will be the distance from point B to the house.  Strangely enough, the distance from point A to the house then is 200 + x - 200, which simplifies to simply x. It's extremely important that you understand that.  

The angle of elevation at point A is 32, and the angle of elevation from point B is 42.  Regardless of the angle in question, the side opposite that angle is the height of the house, y.  

From point A, the distance to the house is the side adjacent to the angle, and the height of the house is the side opposite the angle.  That is the tangent ratio.  Setting up the ratio using the 32 degree angle as the reference angle:

which we remember from above simplifies to

Doesn't tell us much of anything at all.  But we have another angle, 42:

That doesn't tell us much either!  Yikes!! BUT...

let's simplify both of the tangent ratios.  The first one simplifies to

x tan(32) = y  and the second one simplifies to

(x - 200) tan(42) = y

Now that we have both of them solved for y, and that y is the same y cuz it's the height of the same house, we can set y equal to y in terms of x:

(x - 200) tan(42) = x tan(32)

Distribute through the parenthesis to get

x tan(42) - 200tan(42) = x tan(32)

200tan(42) has a value you can find on your calculator, so we will find it and sub it in:

x tan(42) - 180.0 = x tan(32)

Getting the tangent expressions on one side:

-180.0 = x tan(32) - x tan(42)

and factoring out the the x:

-180.0 = x(tan(32) - tan(42)) and simplifying a bit:

-180.0 = x(-.2755346)

and dividing to get x alone, we get that

x = 653 feet.  But that's not the height of the house.  We have to back-substitute in for x to find y:

In x tan(32) = y,

653 tan(32) = y so

y = 408 feet

That's a really tall house, but I checked and double-checked the angles and solved for various things and it's correct!  That was quite an ordeal!

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