Qfyilyi

I'm going into Machine Learning and am currently brushing up on some Calculus on Coursera. Everything was going smoothly until I got to this word problem:

A ladder rests against a wall. The top of the ladder touches the wall at height $12$ meters. The length of the ladder is $4$ meters longer than the distance from the base of the ladder to the wall. Find the length of the ladder.

I am confused as to how to deduce the hypotenuse from the information given above. And have sat here trying different things with no success. What am I missing?

Let $d$ be the distance from the ladder to the wall, and $l$ the length of the ladder.
Then

$l = d + 4; tag 1$

since the wall is mos' likely perpendicular to the ground, we may deploy the Pythagorean theorem and write

$l^2 = (12)^2 + d^2; tag 2$

substituting (1) into (2) yields

$(d + 4)^2 = 144 + d^2, tag 3$

$d^2 + 8d + 16 = 144 + d^2, tag 4$

$8d + 16 = 144 Longrightarrow 8d = 128 Longrightarrow d = 16M Longrightarrow l = 20M. tag 5$

Given the length of the wall as $12$.

Take the length of the base as $x$.

Since, the length of the ladder is $4$ times greater than the base we have $x+4$

Now according to the pythagorean theorem we have,

$$(x+4)^2=12^2+x^2$$
$$x^2+16+8x=144+x^2$$
$$8x=128$$
$$x=16$$

So, the length of the ladder $=x+4implies 16+4=20$