What is the rate of change of the height of the top of the ladder?

[Jan Hill]

Homework Statement

A 10' ladderis leaning against a house when its base starts to slide away
By the time the base is 6' fromthe house, the base is moving away at a rate of 16 ft/sec

a)What is the rate of change of the height of the top of the ladder?

b)At what rate is the area of the triangle formed by the ladder, wall and ground changing?

c)At what rate is the angle between the ladder and the ground changing?

Homework Equations

The Attempt at a Solution

We can figure out the height of the house where the ladder hits it as 8 using the pythagorean theorem
We can let the hypotneuse be s
We need to find dx/dt and we have 4 of the 6 necessary numbers to do that. But to find the unknown, we need 5 of the 6 numbers

We have x, y and s and dx/dt = 16 ft/sec but we need to find dy/dt and for that we need ds/dt but how do we get that?

 

[stevenb]

we need ds/dt but how do we get that?

Wouldn't ds/dt be zero? The ladder is of fixed length, so the hypotenuse will not change in time.

 

[eliya]

You need to set up an equation that represents the height in terms of the distance of the base from the wall and the length of the ladder. Here's I would have solved a:

The height of the ladder at any given time will be the sqrt(l^2 - x^2). Where l is the length of the ladder and x is the distance of the base from the wall. Then you need to take the derivative of that (because you need to find the rate of change of the height) and plug in the known values (the distance of the base from the wall at a given time, and dx/dt).

Parts b and c are solved similarly by forming equations of the quantity you're looking for (although what you're looking for really is the rate of change) in terms of things you already know.

 

[Jan Hill]

What kind of formula can I use for rate of change of the angle?

 

[eliya]

Try to think what is the "thing" that connects between sides of a triangle and angles.