Question

practice jais professor is doing another experiment. this one involves madagascar hissing cockroaches. jai needs to record the amount of oxygen the cockroaches consume as the temperature increases in their terrarium.

temperature (degrees celsius)	oxygen consumed (microliters per gram per hour)
15	140
20	220
25	325

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: Identify variables

Let \(x\) be temperature and \(y\) be oxygen - consumed.

Step2: Sketch the graph (a)

Plot the points \((10,85)\), \((15,140)\), \((20,220)\), \((25,325)\) on a coordinate - plane where the \(x\) - axis represents temperature (in degrees Celsius) and the \(y\) - axis represents oxygen consumed (in microliters per gram per hour).

Step3: Compute first differences (b)

The first differences of \(y\) values:
For \(x = 10\) to \(x = 15\): \(140 - 85=55\)
For \(x = 15\) to \(x = 20\): \(220 - 140 = 80\)
For \(x = 20\) to \(x = 25\): \(325 - 220=105\)
The first differences of \(x\) values are constant (\(15 - 10=20 - 15 = 25 - 20 = 5\)).
The second differences of \(y\) values: \(80 - 55 = 25\) and \(105 - 80 = 25\).

Step4: Determine the pattern (c)

Since the first differences of \(x\) are constant and the second differences of \(y\) are constant, the data show a quadratic pattern. A linear pattern has constant first - differences of \(y\) and an exponential pattern has a constant ratio of consecutive \(y\) values.

Answer:

a. Sketch the points \((10,85)\), \((15,140)\), \((20,220)\), \((25,325)\) on a graph with \(x\) - axis for temperature and \(y\) - axis for oxygen consumed.
b. First differences of \(y\): \(55\), \(80\), \(105\); Second differences of \(y\): \(25\), \(25\)
c. The data show a quadratic pattern because the first differences of \(x\) are constant and the second differences of \(y\) are constant.