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!