Answer:
Step-by-step explanation:
Alright, lets get started.
Please refer the diagram I have attached.
The ladder is represented with red color and the length of the ladder is given as 15 feet.
When the bottom of the ladder is 9 feet from the wall, at that instant, the height of the ladder can be find by using Pythagorean theorem.
[tex]x^2 + y^2 = L^2[/tex]
[tex]9^2 + y^2 = 15^2[/tex]
[tex]81 + y^2 = 225[/tex]
Subtracting 81 in both sides
[tex]y^2 =225-81 = 144[/tex]
Taking square root of both sides
[tex]y = 12[/tex]
Now, we know
[tex]x^2 + y^2 = 15^2[/tex]
Differentiating with respect of t
[tex]\frac{d}{dt}(x^2) +\frac{d}{dt}(y^2) = 0[/tex]
[tex]2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0[/tex]
As per given in question, top of the ladder slides down the wall at the rate of 0.33 ft per second.
Downwards means its negative, so
Plugging the value of [tex]\frac{dy}{dt}[/tex] as -0.33
[tex]2x\frac{dx}{dt} - 2y*0.33= 0[/tex]
[tex]x\frac{dx}{dt} - y*0.33= 0[/tex]
Plugging the value of x and y that we previously found
[tex]9*\frac{dx}{dt}-12*0.33 = 0[/tex]
[tex]9*\frac{dx}{dt}-3.96=0[/tex]
[tex]9*\frac{dx}{dt}=3.96[/tex]
Dividing 9 in both sides
[tex]\frac{dx}{dt}=\frac{3.96}{9}=0.44[/tex]
Hence the bottom of the ladder is sliding at 0.44 feet per second. : Answer
Hope it will help :)
