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. Â 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. Â 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. Â 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. Â
The angle of elevation at point A is 32, and the angle of elevation from point B is 42. Â Regardless of the angle in question, the side opposite that angle is the height of the house, y. Â
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. Â That is the tangent ratio. Â 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. Â But we have another angle, 42:
That doesn't tell us much either! Â Yikes!! BUT...
let's simplify both of the tangent ratios. Â 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. Â But that's not the height of the house. Â 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! Â That was quite an ordeal!