1. ## Trig Differential?

Not sure if this is right forum, but hopefully is.

The height, h centimeters, of a bicycle pedal above the ground at time t seconds is given by the equation:
h=30+15cos90t°

(a)
Calculate the maximum and minimum heights of the pedal
(b) Find the first two positive values of t for which the height of the pedal is 43 cm.
(c) How many revolutions does the pedal make in one minute?

To calculate the max and min, the differential would equal 0 and to determine
if the point is a max or a min, the second differential would be positive or neg?

I know that the differential of cos is -sin, but how do I differentiate something like this?
Also, how do I go about 'c' for that matter?

2. ## Re: Trig Differential?

Hello, witri!

The height, $h$ centimeters, of a bicycle pedal above the ground
at time $t$ seconds is given by the equation: . $h\:=\:30+15\cos90t^o$

(a) Calculate the maximum and minimum heights of the pedal
(b) Find the first two positive values of t for which the height of the pedal is 43 cm.
(c) How many revolutions does the pedal make in one minute?

First, convert degrees to radians: . $h \:=\:30 + 15\cos\left(\tfrac{\pi}{2}t\right)$

Can you finish the problem now?

3. ## Re: Trig Differential?

No, sorry. I have thought about it, but am puzzled still.

4. ## Re: Trig Differential?

Hello, witri

The height, h centimeters, of a bicycle pedal above the ground
at time t seconds is given by the equation: . $h\:=\:30+15\cos\left(\tfrac{\pi}{2}t\right)$

(a) Calculate the maximum and minimum heights of the pedal.

$h' \:=\:\tfrac{15}{2}\sin(\tfrac{\pi}{2}t) \;=\;0 \quad\Rightarrow\quad \sin(\tfrac{\pi}{2}t) \:=\:0$

. . $\tfrac{\pi}{2}t \;=\;0,\,2\pi,\,3\pi,\,4\pi,\,\hdots \quad\Rightarrow\quad t \:=\:0,\,2,\,4,\,6,\hdots$

$h(0) \;=\;30 + 15\cos(0) \;=\; 30 + 15 \;=\;45\text{ cm }\hdots \text{ max}$

$h(2) \;=\;30 + 15\cos(\pi) \;=\;30 - 15 \;=\;15\text{ cm } \hdots \text{ min}$

(b) Find the first two positive values of t for which the height of the pedal is 43 cm.

We want: . $h \:=\:43.$

$30 + 15\cos(\tfrac{\pi}{2}t) \:=\:43 \quad\Rightarrow\quad \cos(\tfrac{\pi}{2}t) \:=\:\tfrac{13}{15} \quad\Rightarrow\quad \tfrac{\pi}{2}t \:=\:\cos^{-1}(\tfrac{13}{15})$

$\tfrac{\pi}{2}t \:=\:\begin{Bmatrix}0.522314822 \\ 5.760870485\end{Bmatrix} \quad\Rightarrow\quad t \;=\;\begin{Bmatrix}0.332515943 \\ 3.667484057\end{Bmatrix}$

Therefore: . $t \;\approx\;\begin{Bmatrix}0.333 \\ 3.667\end{Bmatrix}\:\text{ seconds}$

(c) How many revolutions does the pedal make in one minute?

$\begin{array}{cccccccc}h(0) &=& 30 + 15\cos(0) &=& 30 + 15 & =& 45 \\ h(1) &=& 30 + 15\cos(\frac{\pi}{2}) &=& 30 + 0\; &=& 30 \\ h(2) &=& 30 + 15\cos(\pi) &=& 30 - 15 &=& 15 \\ h(3) &=& 30 + 15\cos(\frac{3\pi}{2}) &=& 30 + 0\; &=& 30 \\ h(4) &=& 30 + 15\cos(2\pi) &=& 30 + 15 &=& 45 \end{array}$

So it takes 4 seconds to make one revolution.

That's 15 revolutions in one minute.

5. ## Re: Trig Differential?

There is no calculus required for c. There are 60 seconds in one minute so "90 t" is 90*60 degrees. A complete circle, one "rotation", has 360= 90*4 degrees. How many complete circles are there in 90*60 degrees?