# Rate of Change Problem with Clock Hand Position

• Mr. Heretic
In summary, the conversation discusses a problem involving a clock with a certain length for the minute hand and hour hand. The question is asking for the rate of change in distance between the tips of the hands at 9 am. The conversation also mentions two different methods for solving the problem, with one method giving an incorrect answer due to a calculation error. Eventually, the correct answer is found using the other method.
Mr. Heretic

## Homework Statement

"On a certain clock, the minute hand is 8 cm long and the hour hand is 6 cm long.
How fast, in cm/min, is the distance between the tips of the hands changing
at 9 am?"

## The Attempt at a Solution

I should be able to transform this somewhat difficult differentiation problem into a simpler limits problem due to the fact that the hands are perpendicular to each other at 9AM, and the velocity of the tip of a hand is perpendicular to the length of the hand at any time.
So where the speed of the tip of the minute hand is 'm' and that of the hour hand is 'h': (sqrt((8 - ht)^2 + (6 + mt)^2) - sqrt(8^2 + 6^2))/t should be an approximation of the change in distance between the tips over period t while t is non-zero, and the limit as t goes to zero should be exactly the rate of change that's desired.
However with m as 4pi/15 cm/min and h as pi/120 cm/min, I get 23pi/150 where the answers state 22pi/150. I've been over my working for an algebra error enough times to be pretty sure there hasn't been one.

The answers use a completely different method which I do understand (cosine rule and chain rule), I just want to know why this method isn't working.

Oh my god, I can't believe it was that... I accidentally did one rotation of the hour hand per 24 hours instead of per 12. Now I'm getting the right answer and I can know my side-step of a method is valid. Thanks, Voko.

## 1. What is a rate of change problem with clock hand position?

A rate of change problem with clock hand position is a mathematical question that involves determining how the position of a clock hand changes over time. This type of problem typically requires knowledge of basic geometry and algebra.

## 2. How do you solve a rate of change problem with clock hand position?

The first step in solving a rate of change problem with clock hand position is to identify the given information and what is being asked for. Then, use the formula "rate = change in position / change in time" to find the rate of change. Finally, use this rate to solve for the unknown variable using algebraic equations.

## 3. What are the units used for measuring the rate of change in clock hand position?

The units used for measuring the rate of change in clock hand position depend on the units used for measuring time and position. For example, if time is measured in seconds and position is measured in degrees, then the rate of change would be in degrees per second.

## 4. How can I apply rate of change problems with clock hand position in real-life situations?

Rate of change problems with clock hand position can be applied in various real-life situations, such as calculating the speed of a rotating object, determining the rate at which the sun moves across the sky, or predicting the position of a planet in its orbit at a certain time. These types of problems can also be used in fields such as engineering, astronomy, and physics.

## 5. What are some tips for solving rate of change problems with clock hand position?

Some tips for solving rate of change problems with clock hand position include carefully reading the problem and identifying the given information, using a diagram or drawing to visualize the situation, and checking the units of measurement to ensure they are consistent. It can also be helpful to break the problem into smaller, easier-to-solve steps.

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