Question

A 10 ft ladder leans against a wall at an angle theta. 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 theta when theta = 60 degrees. Express the answer in units feet/degree.

Answer in book is = 0.087 ft/deg

Answer 1

Draw a diagram, labeling everything. If the top of the ladder is x feet above the ground, how is x related to the length of the ladder (10 feet) and to the angle, theta?

Answer 2

Write a static equation for that (nothing mov

Answer 3

nothing moving).

Answer 4

Now differentiate your equation. Your goal is to find d(theta) / dt, where t = time.

Answer 5

What would you do first?

Answer 6

\[\sin \theta = x / 10\] I have this.

Answer 7

now do I multiply both sides by ten and derive?

Answer 8

The answer in the back of the book is 0.087 ft/deg

Answer 9

What specifically are you looking for? Describe it in words, please.

Answer 10

The rate at which x changes with respect to theta. We are assuming that the ladder bottom of the ladder is being pushed towards the wall.

Answer 11

all right. How would you express that quantity in symbols?
It's a derivative.

Answer 12

\[dx / d\]

Answer 13

wait dx/ d(theta)

Answer 14

The rate at which x changes with respect to theta.

Which variable depends upon which, in this sit'n?

Answer 15

The dependent variable goes on top and the ind..t variable on the bottom when you write this rate of change.

Answer 16

So then we write it as d(theta) / d(x) .

Answer 17

hmm I"d think more like \(\large \left|\cfrac{dx}{d\theta}\right|_{\theta=60^o}\)

Answer 18

That's the reverse: How fast does the angle change with x? We want, how fast does x change with the angle, theta?

Answer 19

We need to find the derivative, with respect to theta, of what relationship?

Answer 20

The relationship between the length of the ladder and x

Answer 21

Yes, and what is that relationship?

Answer 22

sin x /10 or csc 10 / x

Answer 23

But where's theta in those?

Answer 24

Right: think in terms of a trig function.

Answer 25

whoops \[\sin \theta = x / 10\] or \[\csc \theta = 10 / x\]

Answer 26

sin theta = x/10 is correct; the hypo is 10 and x is the height of the top of the ladder abo ve the ground.

Answer 27

You could certainly solve for theta if you wish, altho it's not necessary.

Want to work with sin theta or inverse-sine theta? Your choice.

Answer 28

sin theta

Answer 29

sin theta = x/ 10.

We want to find the rate of change of x with respect to theta when theta = 60 degrees.
Find the derivative, with respect to angle theta, of sin theta = x/10.

Answer 30

10cos(theta) = x

Answer 31

but what about the Chain Rule? recall that x supposedly depends upon theta.

Answer 32

This is where Im stuck. The next section is the chain rule. We haven't covered that yet. But I don't think we can solve this without that rule.

Answer 33

How would you express "the derivative of x with respect to theta?"

Answer 34

Use symbols. Evidently you've heard of "chain rule," even tho' you've not formally studied it.

Answer 35

Find the rate at which x changes with respect to theta when theta = 60 degrees.

Answer 36

sin theta = x / 10.
taking the derivative with respect to theta, we get (1

cos theta = (1/10) (dx/d-theta). Right?

Answer 37

yes

Answer 38

Now let theta = 60 deg. What is cos theta?

Answer 39

1/2

Answer 40

Yes, and so \[\frac{ 1 }{ 2 }=\frac{ 1 }{10}\frac{ dx }{ d }\]

Answer 41

\[\frac{ 1 }{ 2 }=\frac{ 1 }{10}\frac{ dx }{ d \theta }\]

Answer 42

Now please solve this for dx/dtheta.

Answer 43

dx /dtheta = 5

Answer 44

right. that's 5 what? units of measurement? please refer to the original question.

Answer 45

5 feet per degree

Answer 46

exactly. Congrats, you're done!

Answer 47

Be sure to label your answer with dx/dtheta= ...

Answer 48

thanks for the help!!!

Answer 49

Anyquestions? You're welcome!

Answer 50

yea, that's not the answer in the back of the book. 0.087 ft/deg

Answer 51

That's a huge discrepancy. Let's go thru this again fast.

Is sin theta = x / 10 true?

Answer 52

yes

Answer 53

Take the deriv of both sides: cos theta * dtheta/dtheta = (1/100 dx/dtheta. True or false?

Answer 54

False, it's 1/10

Answer 55

False, it's (1/10) * dx / dtheta.

Answer 56

Mult both sides by 10 to remove the fraction 1/10.

Answer 57

yup. So 10 * 1/2 on the left. And dx / dtheta on the right

Answer 58

Yes. Your result includes the data that theta = 60 degrees.

thus, 5= dx / dtheta.

Darn.

if sin theta = x/10
we could solve for theta:

\[\theta=\sin ^{-1}\frac{ x }{ 10 }\]

Answer 59

Takin gthe deriv. with resp. to theta of both sides, we get what?

Answer 60

dtheta
------ = 1
dtheta

Answer 61

So:\[1=\frac{ 1 }{ ^{\sqrt{1-(x/10)^2}} }\frac{ dx }{d \theta }\]

Answer 62

You'll have to find the value of x when the angle theta is 60 degrees.

Sorry, we must be overlooking some important fact here, if our answer is that much different from that in the book.

Answer 63

Its fine

Answer 64

I have to get off the 'Net now, but would be glad to continue with you later on.
Thanks for your participation and fast responses.

Answer 65

Thanks for the help!!

Answer 66

My pleasure!

Answer 67

I figured out how to get the right answer. The 5 that we got is correct. But you still need to convert that into feet per degree. You do this by multiplying 5 by PI and then dividing by 180.