Functions, Linear Equations and Proportional Figures

On the first day after the new moon, 2% of the moon's surface is illuminated. On the second day, 6% is illuminated.

1. Based on this information, predict the day on which the moon's surface is 50% illuminated and 100% illuminated.
2. The moon's surface is 100% illuminated on day 14. Does this agree with the prediction you made?
3. Is the percentage illumination of the moon's surface a linear function of the day?

When the sun was directly overhead, the stick had no shadow. After 20 minutes, the shadow was 10.5 cm long. After 60 minutes, it was 26 cm long.

1. Based on this information, estimate how long it will be after 95 minutes.
2. After 95 minutes, the shadow measured 38.5 cm. How does this compare to your estimate?
3. Is the length of the shadow a function of time? If so, is it linear? Explain your thinking.

1. Answers vary. Sample response: If we model this with the linear function that goes through (0,0) and (20,10.5), we would predict that the length would be growing at a rate of 10.520 centimeters per minute. After 95 minutes, this would give a prediction of about 50 centimeters since 95⋅10.520≈49.88.
2. Answers vary. Sample response: The prediction we made overestimated the length by about 11.5 centimeters.
3. The length of the shadow is a function of time, since every time determines only one length. It is not a linear function of the time, since the points (0,0),(20,10.5),(60,26), and (95,38.5) do not lie on any one line. Based on the answer to the last part, it is not even very well approximated by a linear function.

A candle is burning. It starts out 12 inches long. After 1 hour, it is 10 inches long. After 3 hours, it is 5.5 inches long.

1. When do you think the candle will burn out completely?
2. Is the height of the candle a function of time? If yes, is it a linear function? Explain your thinking.

In science class, Jada uses a graduated cylinder with water in it to measure the volume of some marbles. After dropping in 4 marbles so they are all under water, the water in the cylinder is at a height of 10 milliliters. After dropping in 6 marbles so they are all under water, the water in the cylinder is at a height of 11 milliliters.

1. What is the volume of 1 marble?
2. How much water was in the cylinder before any marbles were dropped in?
3. What should be the height of the water after 13 marbles are dropped in?
4. Is the volume of water a linear relationship with the number of marbles dropped in the graduated cylinder? If so, what does the slope of the line mean? If not, explain your reasoning.

For a certain city, the high temperatures (in degrees Celsius) are plotted against the number of days after the new year.Based on this information, is the high temperature in this city a linear function of the number of days after the new year?