A 41 foot long ladder
is leaning against a vertical wall of a building.
The base of the ladder is being pulled away from the wall
at a rate of seven feet per second.
Determine the rate of change of the top of the ladder
when the base of the ladder is nine feet away from the wall.
Let's first label all three sides of the right triangle.
Let's label the a horizontal leg x, the vertical leg y,
and the hypothenuse,
which would be the length of the ladder, z.
And we know z, of course, is 41 feet.
We're also told that the base of the ladder
is being pulled away from the wall
at a rate of seven feet per second,
which would be in this direction here.
Notice x is increasing at a rate of seven feet per second,
which we state as dx/dt equals seven feet per second.
We're trying to determine the rate of change
of the top of the ladder.
We'll notice how as x increases,
the top the ladder is going to be moving downward.
We express this rate as dy/dt.
So our goal here is to determine dy/dt
right when the base of the ladder is nine feet away
from the wall, which means when x is equal to nine feet.
And now we need to write an equation that relates x and y,
and we'll also have to include z,
because we'll be using the Pythagorean theorem.
So applying the Pythagorean theorem,
we know that x squared plus y squared is equal to z squared.
But we also know z never changes,
and therefore, we can substitute 41 for z.
This gives us the equation x squared plus y squared
equals the square of 41.
And now we differentiate both sides of the equation
with respect to t.
We have the derivative of x squared plus y squared
with respect to t equals the derivative of 41 squared
with respect to t.
On the left, because we're differentiating
with respect to t and we have x and y terms,
we need to apply the chain rule.
The derivative of x squared the respect to t
is equal to 2x times dx/dt.
Again, we're applying the chain rule here.
Plus the derivative of y squared with respect to t
is 2y times dy/dt.
This is equal to the derivative of 41 squared
with respect to t.
And since 41 squared is a constant, the derivative is zero.
From here, because we're looking for dy/dt,
we now need to solve the equation for dy/dt.
Let's first subtract 2x dx/dt on both sides.
This gives us 2y times dy/dt
equals negative 2x times dx/dt.
And the final step to solve for dy/dt
is to divide both sides by 2y.
Simplifying, we have dy/dt equals,
of course 2 divided by 2 simplifies to 1.
We have negative x times dx/dt,
all divided by y.
Notice, to find dy/dt, we need to know x, y and dx/dt.
We know dx/dt is seven feet per second.
We're trying to find a dy/dt when x equals 9,
but we still have to determine the value of y
when x is 9, and we know Z is 41.
So if x is equal to 9, using the Pythagorean theorem,
we know y squared plus 9 squared
must equal 41 squared.
And therefore, y squared is equal to 41 squared
minus 9 squared.
Or we can say, y is equal to the square root
of the difference of 41 squared and 9 squared,
giving us y equals 40.
And now we can find dy/dt.
Dy/dt is equal to negative x,
which is negative 9 times dx/dt, which is 7,
all divided by y, which we now know is 40.
Simplifying, we have negative 63/40.
And again, this is measuring the change in y
with respect to time in seconds, and y is measured in feet.
The units here are feet per second.
So now we know that change in y with respect to time
is negative 63/40 feet per second,
or y is decreasing at a rate of 63/40 feet per second,
under the given conditions.
I hope you found this helpful.