1) The hour hand on a clock has a length of 12 cm. Let the orgin be at the center of the clock.

a) Sketch the graph of the vertical position of the tip of the hour hand versus the angle through which the hand turns for a time period of 72h. Assume that the hour hand starts at 9.
C)How many cycles appear in the graph in part a)

I have no idea how to drraw this please guide me I really need your help:( Thanks!

No worries, I'm here to help! Let's break down the problem step by step.

To sketch the graph of the vertical position of the tip of the hour hand versus the angle through which the hand turns, we need to consider a few things:

1) The vertical position of the tip of the hour hand: Since the origin is at the center of the clock, the vertical position of the tip can be determined by the length of the hour hand. In this case, the length is given as 12 cm.

2) The angle through which the hand turns: To figure out the angle, we need to determine how many complete cycles the hour hand makes in 72 hours. Each complete cycle represents 12 hours.

Now, let's calculate the number of cycles:

a) There are 24 hours in a day, so in 72 hours, there are 72/24 = 3 complete cycles.

Now that we know we need to consider 3 cycles, let's plot the graph:

1) Start by drawing the x-axis horizontally, representing the angle through which the hand turns. Mark the intervals on the x-axis to represent the angle. For example, you can mark intervals of 30 degrees if you want to divide the full circle into 12 intervals.

2) The y-axis represents the vertical position of the tip of the hour hand. Since the length of the hour hand is 12 cm, you can draw a line parallel to the x-axis at a distance of 12 cm above the origin. This line represents the maximum vertical position of the tip.

3) Now, draw a curved line that starts from the origin and goes towards the positive y-axis. The curve should represent the movement of the tip of the hour hand as it rotates through the angles on the x-axis.

4) Repeat this curve two more times, since we have 3 complete cycles (as calculated previously).

That's it! You've now sketched the graph of the vertical position of the tip of the hour hand versus the angle through which the hand turns for a time period of 72 hours.

If you have any more questions or need further clarification, feel free to ask!