Algebra 2 Problems of the Day (open ended)

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August 2017, Part III

All Questions in Part III are worth 4 credits. Work need be shown (or explained or justified) for full credit. Correct numerical answers with no work receive one credit.


35.a) On the axes below, sketch at least one cycle of a sine curve with an amplitude of 2, a midline at y = -3/2 and a period of 2π.

b) Explain any differences between a sketch of y = 2 sin (x - π/3) - 3/2 and the sketch from part a.

Answer:
The sine curve starts at the midline value. In y = sin x, the midline would be zero, and it would start at the origin. This curve will start at (0, -3/2). The amplitude is 2, so it has a maximum at -3/2 + 2 = 0.5 (or 1/2), and a minimum at -3/2 - 2 = -3.5 (or -7/2). Label the graph!

They didn't ask for it, but you can create the equation from the description above. (This may also be useful if you wish to use your calculator.)
With an amplitude of 2, period of 2π, and midline of y = -3/2, the sine curve would be

y = 2 sin x - 3/2
See the image below.

In part b, the only difference between the original curve and the new curve is the "- π/3" in the parentheses. This represents a translation of the original curve.
The original sketch will shift π/3 units to the right.

36. Using a microscope, a researcher observed and recorded the number of bacteria spores on a large sample of uniformly sized pieces of meat kept at room temperature. A summary of the data she recorded is shown in the table below.

Using these data, write an exponential regression equation, rounding all values to the nearest thousandth.

The researcher knows that people are likely to suffer from food-borne illness if the number of spores exceeds 100. Using the exponential regression equation, determine the maximum amount of time, to the nearest quarter hour, that the meat can be kept at room temperature safely

Answer:
Put the data into lists L1 and L2 in your calculator and do an exponential regression. You will get, to the nearest thousandth:
y = 4.168(3.981)^x

In part b, substitute 100 for y and solve for x.

100 = 4.168(3.981)^x
100/4.168 = (3.981)^x
23.99232 = (3.981)^x
log (23.99232) = log (3.981)^x
log (23.99232) = x log (3.981)
log (23.99232) / log (3.981) = x
x = 2.300...
The nearest quarter hour is 2 1/4 hours or 2 hours and 15 minutes.



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