Question

A ladder is leaning against a building. The distance from the bottom of the ladder to the building is 16 ft less than the length of the ladder. How high up the side of the building is the top of the ladder if that distance is 2 ft less than the length of the​ ladder?

Answer 1

I'm going to set the length of the ladder equal to
`x`.

We can now say that the distance from the bottom of the ladder to the building is 16 less than
`x`, or
`x-16`.

We also know that the distancefrom the height of the top of the ladder is 2 less than x, or
`x-2`. We now have 3 sides:

`a = x-16`

`b = x-2`

`c = x`

We know this is a right triangle (assuming that the building and the ground are at right angles) so we can use pythagorean theorem:

`a^2+b^2=c^2`

Let's plug in our values:

`(x-16)^2+(x-2)^2=x^2`

Let's now expand these using the FOIL method or the fact that
`(x-a)^2=x^2-2ax+a^2`

`x^2-32x+256+x^2-4x+4=x^2`

Now we can combine everything:

`x^2-36x+260=0`

Now we can factor: We're looking for factors of 260 that add to -36. We get -26 and -10.

`(x-26)(x-10)=0`

Now we know
`x=26` or
`x=10`

We know that
`x=26` because one of the sides is 16 less than
`x` and 16 less than 26 is negative, we can't have a negative side.

We're trying to find the height, which we previously said was
`x-2=26-2=24`

The height is 24ft.

Answer 2

Let's say the length of the ladder is "d". Since the distance from the bottom of the ladder to the building is 16 feet less than the length, we can write this as d - 16.

The distance from the ground to the top of the ladder is 2 feet less than the length of the ladder, so we can denote this as

d - 2.

The shape made by the ground, ladder, and side of the building is a right triangle, and we just found its legs. We can use the Pythagorean Theorem to solve for "d" and find how far up the side of the building the ladder is:

(d - 16)^2 + (d - 2)^2 = d^2

(d^2 - 32d + 256) + (d^2 - 4d + 4) = d^2

2d^2 - 36d + 260 = d^2

d^2 - 36d + 260 = 0

(d - 26) * (d - 10) = 0

d = 26 or d = 10

The answer d = 10 does not fit in this case because it isn't possible to have 16 feet less of 10 feet; it would be negative!

So, d = 26. The top of the ladder to the ground is 2 less than d, so we do 26 - 2 = 24 feet.

The answer is 24 feet.