Math word problems

This is simply an application of the pythagorean theorem which states for any right triangle, a2 + b2 = c2 where c is the hypotenuse while a and b are the sides of the triangle.

In this example one side of the triangle is 16ft another (the hypotenuse) is 30ft and the height which the ladder reaches is unknown so we call it X.

Therefore 30squared = 16squared + X Squared

Then subtract 16squared from both sides

30squared - 16squared = Xsquared
Gives us:

900 - 256 = Xsquared
644 = Xsquared

Then take the square root of both sides and

X = square root of 644 or approximately 25.38ft


Take the square of both sides and you get

X= 34ft

Yes, it is a simple pythag problem. Let's give this problem a bit of a twist though. Let us assume that the tip of the ladder is slipping down the wall at a rate of 1 foot per second. How fast is the angle between the ladder and the wall changing when it is 20 feet up the wall?

Are we also assuming the ladder is retracting to allow this slippage to continue?

All 3 angles of a triangle equate to 180 degrees. We know the right angle will never change and always, leaving us with 90 degrees to play with.

The hypotnuse doesn't have a factor in this answer.

The shorter side of the triangle will have the larger angle on its other end.

With side lengths of 20 and 16. The total of the sides are 36.
To figure out what percentage the angle is we just divide 20/36 = 55.5555%
Take our total of 90 degrees and now multiply by .5555555 leaves us with 50.

So one angle is 50 degrees and the other is 40. We don't need to factor the hypotnuse into this because the right angle will never change.

I already said the shorter side gets the larger angle. So the ladder is at a 40 degree angle to the wall since its the larger side.

One second later we apply the same math to 19 feet.
19+16 = 35
19/35 = 54.28571
90*.5428571 = 48.85
90-48.85= 41.15

The difference between the angle at 20 feet and 19 feet is 1.15 degrees

At 20ft the rate of change is about +1.15 degrees per second. But this will get larger as the tip of the ladder gets closer to the ground. I don't remember how to express this. I don't remember a problem like this in middle school.