Question

1. practice jai is also supposed to keep track of the amount of food the lab rats eat each day. the rat food is kept in a large container. a technician feeds the rats from the container. each day jai weighs the food remaining in the large container to see how much the rats ate.

time since start of experiment (days)	amount of food remaining (kg)
2	21.28
3	21.04
4	20.80
5	20.56

a. sketch a graph showing the data in the table.
b. compute the first and second differences.
c. do these data show a linear, quadratic, or exponential pattern? explain.

Explanation:

Step1: Sketch the graph

On the x - axis, label "Time since start of experiment (days)" with values 1, 2, 3, 4, 5. On the y - axis, label "Amount of food remaining (kg)" with an appropriate scale to fit values from 20.56 to 21.52. Plot the points (1, 21.52), (2, 21.28), (3, 21.04), (4, 20.80), (5, 20.56) and connect them with a smooth curve.

Step2: Compute first - differences

Let \(y_i\) be the amount of food remaining at day \(i\). The first - differences \(\Delta y_i=y_{i + 1}-y_i\).
\(\Delta y_1=21.28 - 21.52=- 0.24\)
\(\Delta y_2=21.04 - 21.28=-0.24\)
\(\Delta y_3=20.80 - 21.04=-0.24\)
\(\Delta y_4=20.56 - 20.80=-0.24\)

Step3: Compute second - differences

Since the first - differences are constant (\(\Delta y_i=-0.24\) for \(i = 1,2,3,4\)), the second - differences \(\Delta^2y_i=\Delta y_{i + 1}-\Delta y_i = 0\) for \(i=1,2,3\).

Step4: Determine the pattern

A linear function has constant first - differences and zero second - differences. Since the first - differences are constant (-0.24) and the second - differences are 0, the data shows a linear pattern.

Answer:

a. Graph: Points (1, 21.52), (2, 21.28), (3, 21.04), (4, 20.80), (5, 20.56) plotted and connected with a smooth curve.
b. First - differences: -0.24 for all intervals. Second - differences: 0 for all intervals.
c. The data shows a linear pattern because the first - differences are constant (-0.24) and the second - differences are 0.