Further investigation of classic ladder problem

[LearninDaMath]

Homework Statement

I already know the answer, and know "how" to get the answer to this problem:


A 10 foot ft ladder leans against a wall at an angle θ with the horizontal [ground], as shown in the accompanying figure (the figure is of a ladder leaning against a wall). The top of the ladder is x feet above the ground. If the bottom of the ladder is pushed toward the wall, find the rate at which x changes with respect to θ when θ = 60 degrees. Express the answer in units of feet/degree.

My question is:

I understand that a rate is a derivative. And derivatives are expressed as tangent lines to a function on a graph. So I am wondering how this ladder problem would be expressed on a graph. Could it be expressed on the cartesian coordinate system? What would it look like? What would the x coordinates and y coordinates be?

 

[Ray Vickson]

Homework Statement

I already know the answer, and know "how" to get the answer to this problem:


A 10 foot ft ladder leans against a wall at an angle θ with the horizontal [ground], as shown in the accompanying figure (the figure is of a ladder leaning against a wall). The top of the ladder is x feet above the ground. If the bottom of the ladder is pushed toward the wall, find the rate at which x changes with respect to θ when θ = 60 degrees. Express the answer in units of feet/degree.

My question is:

I understand that a rate is a derivative. And derivatives are expressed as tangent lines to a function on a graph. So I am wondering how this ladder problem would be expressed on a graph. Could it be expressed on the cartesian coordinate system? What would it look like? What would the x coordinates and y coordinates be?

Just figure out what x is as a function of θ.

RGV

 

[LearninDaMath]

Just figure out what x is as a function of θ.

RGV

So for example, if I had a position vs time graph, that means the y-axis would be a position axis and the x-axis would be a time axis.

So are you saying that in this case, the y-axis should be an "angle" axis and the x-axis should be a position axis? So that I have an "angle vs position" graph?

 

[LearninDaMath]

Just figure out what x is as a function of θ.

RGV

I still don't know what you mean. This is a function in terms of θ. Sinθ=x/10, so maybe i don't understand the terminology. What do you mean find out what x is as a function of θ? Are you saying find dx/dθ? I already found that to be 5ft/rad. But in terminology, I would have thought it would be said like this: "find derivative of x in terms of θ" ...so is that the same thing as saying: "find x as a function of θ"?

 

[LearninDaMath]

So you're saying I could represent it on trig graph like this?:

http://en.wikipedia.org/wiki/File:Sine.svg

Where the horizontal axis would be the angles and the vertical axis would be the height? So then the derivative of sin(60) would be cos60, or 1/2? So the slope or rate is 5?

 

[Ray Vickson]

I still don't know what you mean. This is a function in terms of θ. Sinθ=x/10, so maybe i don't understand the terminology. What do you mean find out what x is as a function of θ? Are you saying find dx/dθ? I already found that to be 5ft/rad. But in terminology, I would have thought it would be said like this: "find derivative of x in terms of θ" ...so is that the same thing as saying: "find x as a function of θ"?

For every θ between 1 and π/2 you can figure out what x must be to match that θ. So, YES, you get a function x = f(θ), and its derivative df/dθ give you exactly what the question asks for, if you go back and read it again.

RGV

 

[LearninDaMath]

Right, but I already got the answer that the question asks for...I know that its 5ft/rad. I'm not confused about how to solve this question for the correct answer.


My question is, how is that represented on a coordinate system? (that is not part of any assigned question)

Which coordinate system should I use? (that is not part of any assigned question)

Where are you getting n/2 from? (that is not part of any assigned question)

Its just wondering how a scenario of a ladder moving would look if graphed. I can't picture it.

Am I on the right track with the sin graph?