Answer: The ladder is sliding down the wall at a rate of [tex]5\dfrac{17}{50}\ ft/sec[/tex]
Step-by-step explanation:
Since we have given that
Length of ladder = 25 foot
Distance from the wall to the bottom of ladder = 15 feet
Let base be 'x'.
Let length of wall be 'y'.
So, by pythagorus theorem, we get that
[tex]x^2+y^2=25^2\\\\15^2+y^2+625\\\\225+y^2=625\\\\y^2=625-225\\\\y^2=400\\\\y=\sqrt{400}\\\\y=20\ feet[/tex]
[tex]\dfrac{dy}{dx}=-4\ ft/sec[/tex]
Now, the equation would be
[tex]x^2+y^2=625\\[/tex]
Differentiating w.r.t x, we get that
[tex]2x\dfrac{dx}{dt}+2y.\dfrac{dy}{dt}=0\\\\2\times 15\dfrac{dx}{dt}+2\times 20\times -4=0\\\\30\dfrac{dx}{dt}-160=0\\\\\dfrac{dx}{dt}=\dfrac{160}{30}=5.34\ ft/sec[/tex]
Hence, the ladder is sliding down the wall at a rate of [tex]5\dfrac{17}{50}\ ft/sec[/tex]
