HELP! Rate of change....

[nikchic5]

An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle theta with the plane, then the magnitude of the force is

F= (kW) / (k sin (theta) + cos (theta))

where k is a constant called the coefficient of friction.

(a) Find the rate of change of F with respect to theta

(b) When is this rate of change equal to 0?

(c) If W= 40 lb. and k= 0.7, draw the graph of F as a function of theta and use it to locate the value of theta for which (dF) / (d(theta)) = 0.

If anyone could help me I would really appreciate it!! Thanks!

 

[soroban]

Hello, nikchic5!

Exactly where is your difficulty?
$\displaystyle \;\;$You don't know that a rate-of-change is a derivative?
$\displaystyle \;\;$You don't know how to differentiate that function?
$\displaystyle \;\;$You can't set it equal to zero and solve for $\displaystyle \theta$ ?

An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object.
If the rope makes an angle $\displaystyle \theta$ with the plane, then the magnitude of the force is:
$\displaystyle \;\;\;F\;=\;\frac{kW}{k\sin\theta\,+\,\cos\theta}$

where $\displaystyle k$ is a constant called the coefficient of friction.

(a) Find the rate of change of $\displaystyle F$ with respect to $\displaystyle \theta$.

(b) When is this rate of change equal to 0?

(c) If W= 40 lb. and k= 0.7, draw the graph of $\displaystyle F$ as a function of $\displaystyle \theta$
and use it to locate the value of $\displaystyle \theta$ for which: $\displaystyle \,\frac{dF}{d\theta}\,=\,0$

(a) We have: $\displaystyle \,F\;=\;kW(k\cdot\sin\theta\,+\,\cos\theta)^{-1}$

Then: $\displaystyle \,\frac{dF}{d\theta}\;= \;-kW(k\cdot\sin\theta\,+\,\cos\theta)^{-2}(k\cdot\cos\theta\,-\,\sin\theta)\;=\;\frac{kW(\sin\theta\,-\,k\cdot\cos\theta)}{(k\cdot\sin\theta\,+\,\cos\theta)^2}$


(b) If $\displaystyle \,\frac{dF}{d\theta}\,=\,0,\,$ then: $\displaystyle \,\sin\theta\,-\,k\cdot\cos\theta\:=\:0\;\;\Rightarrow\;\;\sin\theta\:=\:k\cdot\cos\theta$

Then: $\displaystyle \,\frac{\sin\theta}{\cos\theta}\:=\:k\;\;\Rightarrow\;\;\theta\:=\:\arctan(k)$


(c) I'll let you do the graphing . . .

The function is: $\displaystyle \:F\;=\;\frac{28}{0.7\sin\theta\,+\,\cos\theta}$

The critical value occurs at: $\displaystyle \,\theta\:=\:\arctan(0.7)\:=\:0.610725974\,$radians $\displaystyle \,\approx\: 35^o$

 

[nikchic5]

thank you very much you were very helpful!