Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2013

Part III: Each correct answer will receive 4 credits. Partial credit is possible.


36. Solve algebraically for all values of x:

log_{(x + 4)}(17x - 4) = 2

Answer:

Telling you to answer in simplest radical form is a sure sign that you need to use the Quadratic Formula.

log_{(x + 4)}(17x - 4) = 2

(x + 4)² = 17x - 4

x² + 8x + 16 = 17x - 4

x² - 9x + 20 = 0

(x - 5)(x - 4) = 0

x - 5 = 0 or x - 4 = 0
x = 5 or x = 4

37. The data collected by a biologist showing the growth of a colony of bacteria at the end of each hour are displayed in the table below.

Write an exponential regression equation to model these data. Round all values to the nearest thousandth.
Assuming this trend continues, use this equation to estimate, to the nearest ten, the number of bacteria in the colony at the end of 7 hours.

Answer:

Put the numbers in a List in your graphing calculator and do an exponential regression.

You should get the following: a = 215.953, b = 1.652, and y = 215.953 * 1.652^x

If the trend continues for 7 hours, then for x = 7, y = 215.953*1.652⁷ = 7251.52824, which is 7250, to the nearest ten.

Note that it said "nearest ten" not "nearest tenth", which doesn't make sense for a bacteria colony.

38. As shown in the diagram below, fire-tracking station A is 100 miles due west of fire-tracking station B. A forest fire is spotted at F, on a bearing 47° northeast of station A and 15° northeast of station B. Determine, to the nearest tenth of a mile, the distance the fire is from both station A and station B. [N represents due north.]

Answer:

Label what you know about the triangle, and then use the Law of Sines.

We know what c is (or in this case, f), so we can use that to find a and b.

sin A / a = sin C / c
sin 43 / a = sin 32 / 100
a = 100 * sin 43 / sin 32 = 128.6985...
The distance from station B to the fire is 128.7 miles.

sin B / b = sin C / c
sin 105 / a = sin 32 / 100
a = 100 * sin 105 / sin 32 = 182.2779...
The distance from station A to the fire is 182.3 miles.

End of Part III.

More to come. Comments and questions welcome.

More Regents problems.

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