A 10 ft ladder leans against a wall. The bottom slides away at 2 ft/s. How fast is the top sliding down when the bottom is 6 ft from the wall?

Problem: A $10\text{ ft}$ ladder leans against a wall, with the bottom sliding away at $2\text{ ft/s}$. Find the rate at which the top is sliding down when the bottom is $6\text{ ft}$ from the wall.

Steps

1. Define variables and write the Pythagorean theorem
   $$x^2 + y^2 = 10^2$$
2. Differentiate both sides with respect to time $t$
   $$2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0$$
3. Find the height $y$ when $x = 6$
   $$6^2 + y^2 = 100$$
4. Simplify the equation for $y^2$
   $$36 + y^2 = 100$$
5. Subtract 36 from both sides
   $$y^2 = 64$$
6. Take the positive square root for the height
   $$y = 8$$
7. Substitute known values ($x = 6$, $y = 8$, $\frac{dx}{dt} = 2$) into the derivative equation
   $$2(6)(2) + 2(8)\frac{dy}{dt} = 0$$
8. Simplify the coefficients
   $$24 + 16\frac{dy}{dt} = 0$$
9. Subtract 24 from both sides
   $$16\frac{dy}{dt} = -24$$
10. Divide both sides by 16
    $$\frac{dy}{dt} = -\frac{24}{16}$$
11. Reduce the fraction
    $$\frac{dy}{dt} = -\frac{3}{2}$$