Related Rate

[jbauman793]

At what rate is the distance between the tip of the second hand and the 12 o'clock mark changing when the second hand points to 4 o'clock?

This is a Related Rate Problem...and i have no idea how to start..please help.

 

[soroban]

Hello, bauman793

At what rate is the distance between the tip of the second hand and the 12 o'clock mark changing when the second hand points to 4:00?

The answer depends on \(\displaystyle R\),the radius of the clock face.
. . Having said that, I'll continue . . .


I assume that you refer to the Hour hand, the Minute, and the Second hand.

Since the Second hand is the longest hand, I further assume that the length of the Second hand is the radius of the clock.

                A
              * o *
          *     *     *
                *       *
       *      R *        *
                *
      *         *         *
      *         o  @      *
      *        O  *       *
                    *
       *          R   *  *
        *               o
          *           *   B
              * * *

The center of the clock is at \(\displaystyle O,\)
The 12:00 mark is at \(\displaystyle A.\)
The tip of the Second hand is at \(\displaystyle B.\)
Let \(\displaystyle OA \,=\, OB \,=\, R\text{ inches.}\)

Draw \(\displaystyle AB.\) .Let \(\displaystyle x \:=\:AB.\)
Let \(\displaystyle \theta = \angle AOB.\)


\(\displaystyle \text{Law of Cosines: }\:x^2 \;=\;R^2 + R^2 - 2(R)(R)\cos\theta \;=\;2R^2 - 2r^2\cos\theta \;=\;2R^2(1-\cos\theta) \;=\;4R^2\left(\frac{1-\cos\theta}{2}\right) \;=\;4R^2\sin^2\left(\tfrac{\theta}{2}\right)\)

. . \(\displaystyle \text{Hence: }\:x \;=\;2R\sin\left(\tfrac{\theta}{2}\right)\)

\(\displaystyle \text{Differentiate with respect to time: }\;\frac{dx}{dt} \;=\;R\cos\tfrac{\theta}{2}\,d\theta\)


\(\displaystyle \text{At 4:00, }\theta \,=\,\frac{2\pi}{3}\)

\(\displaystyle \text{Also: }\:\frac{d\theta}{dt} \:=\:\frac{2\pi\text{ radians}}{60\text{ seconds}} \:=\:\frac{\pi}{30}\text{ rad/sec}\)


\(\displaystyle \text{Therefore: }\:\frac{dx}{dt} \;=\;R\cos\tfrac{\pi}{3}\cdot\tfrac{\pi}{30} \;=\; R\cdot\tfrac{1}{2}\cdot\tfrac{\pi}{30} \;=\;\frac{\pi}{60}R \text{ inches/second.}\)