A 6 ft ladder, resting against a wall, begins to slip down the wall. When the angle of the ladder is 45 degrees, the bottom of the ladder is moving away from the wall at 0.5 m/s. At that moment, how fast is the top of ladder moving down the wall?

Step-by-step explanation:

This is a related rates problem from calculus using implicit differentiation. The main equation is going to be Pythagorean's Theorem and then the derivative of that. Pythagorean's Theorem is

where c is the hypotenuse and is a constant. Therefore, the derivative of this with respect to time, and using implicit differentiation is

and dividing everything by 2 to simplify a bit:

. Upon analyzing that equation, it looks like we need values for x, y, , and . And here's what we were given:

and  In the greater realm of things, that's nothing at all.

BUT we can use the right triangle and the angle we were given to find both x and y. The problem we are looking to solve is to

Find at the instant that = .5.

Solving for x and y:

and

6tan45 = x ( and since this is a 45-45-90 triangle, y = x):

so

and now we can fill in our derivative. Remember the derivative was found to be

so

and

and

and multiplying by the reciprocal of the left gives us:

so

thats not a system of equations, that is a solution