What shadow would a 137cm tall child cast at 9am 10am 11am 12noon 1pm 2pm 3pm and 4pm if they were facing north in january?

There are so many variables undefined that it is impossible to say.

Are there clouds?
Is the ground level?
Is the child standing straight up? sitting? lying down?
What latitude is involved?

What possible difference does it make which direction the child is facing?

Who are "they"?

Some of the questions may seem ridiculous (and some are intended to be), but it shows how important it is to nail down the specifics of a situation before asking questions about it.

That would depend upon the latitude, since it determines the sun's elevation at those hours.

Also, the child's shadow's length will not depend upon the direction he or she is facing.

Who thought up this crazy question? Take a look at whether your method of

..whether your method of education is appropriate

To determine the shadow length, we need to consider the position of the sun throughout the day and the direction the child is facing.

In January, the sun is typically in the southern part of the sky (assuming you're in the northern hemisphere). Since the child is facing north, the shadow will be cast in the opposite direction, to the south. This means that their shadow will be on the ground behind them.

To estimate the length of the shadow, we need to know the angle of the sun at different times of the day. Since the child's height is given as 137cm, we can assume their shadow will be proportional to their height.

Let's calculate the approximate shadow lengths for each hour:

1. 9am: The sun is still relatively low in the sky, so the shadow will be longer. Assuming a 45-degree angle between the sun and the ground, the child's shadow will be roughly equal to their height, which is 137cm.

2. 10am: The sun rises further, but the angle is still relatively low. The shadow length will increase but not drastically. Assuming a 35-degree angle, the shadow length might be around 2/3 of the child's height, which is approximately 91cm.

3. 11am: The sun is higher in the sky compared to earlier hours. The shadow length will decrease accordingly. Assuming a 25-degree angle, the shadow length might be around 1/3 of the child's height, which is approximately 45cm.

4. 12noon: At noon, the sun is at its highest point, creating almost no shadow directly beneath an object. The shadow length will be very minimal, almost negligible.

5. 1pm: After noon, the sun starts declining in the sky. The shadow length will gradually increase again. Assuming a 25-degree angle (same as 11am), the shadow length might be around 1/3 of the child's height, which is approximately 45cm.

6. 2pm: The sun is lower than 1pm but still fairly high. The shadow length will be slightly longer. Assuming a 35-degree angle, the shadow length might be around 2/3 of the child's height, which is approximately 91cm.

7. 3pm: The sun continues to descend. The shadow length will increase. Assuming a 45-degree angle (same as 9am), the shadow length might be roughly equal to the child's height, which is 137cm.

8. 4pm: The sun is getting close to the horizon. The shadow will be longer. Assuming a 55-degree angle, the shadow length might be around 4/3 of the child's height, which is approximately 183cm.

Keep in mind that these estimations assume a clear sky and approximate values for the sun's angle. The actual lengths can vary depending on various factors such as geographic location, weather conditions, and atmospheric conditions.