Answer:
[tex]\dot y = -1.667\,\frac{ft}{s}[/tex]
Explanation:
Let consider positive when top of the ladders slides up and the bottom of the ladder slides away from the wall. Ladder can be described by Pythagorean Theorem:
[tex]r^{2} = x^{2} + y^{2}[/tex]
Where:
[tex]r[/tex] is the ladder length, [tex]x[/tex] is the horizontal distance between the bottom of the ladder and wall and [tex]y[/tex] is the vertical distance between floor and the top of the ladder.
An expression describing velocities at top and bottom of the ladder can be obtained by using derivatives as a function of time:
[tex]2\cdot x \cdot \dot x\, + 2\cdot y \cdot \dot y = 0[/tex]
[tex]x\cdot \dot x\, + y \cdot \dot y = 0[/tex]
The rate for the top of the ladder is:
[tex]\dot y = - \frac{x}{y}\cdot \dot x[/tex]
[tex]\dot y = -\frac{10\,ft}{\sqrt{(26\,ft)^{2}-(10\,ft)^{2}} } \cdot (4\,\frac{ft}{s} )[/tex]
[tex]\dot y = -1.667\,\frac{ft}{s}[/tex]
