The top of a ladder slides down a vertical wall at a rate of 0.375 m/s. at the moment when the bottom of the ladder is 6 m from the wall, it slides away from the wall at a rate of 0.5 m/s. how long is the ladder?

The ladder is moving away from the wall at the bottom, namely dx/dt
at a rate of 0.5 m/s, now... when it's 6m away from the wall, it has been moving for 12seconds already then, at 0.5ms, you need 12s to get 6meters away, thus x = 6

the ladder is moving downwards, namely dy/dt
at a rate of 0.375 m/s, in 12seconds, it has moved then 4.5meters, thus y = 4.5

so hmmm, notice, the ladder is sliding down and away from the wall, but the ladder itself itself elongating or shrinking, so is a constant

let us use the pythagorean theorem for that, notice picture below
thus



pretty sure is very obvious :)

When rates are expressed as a quantity of 1, such as 2 feet per second or 5 miles per hour, they are called unit rates. if you have a multiple-unit rate such as 120 students for every 3 buses, and want to find the single-unit rate, write a ratio equal to the multiple-unit rate with 1 as the second term.