Algebra 2 Problems of the Day (Algebra 2 Regents, June 2012)

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, June 2012

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


36. Express in the simplest form:

( (4 - x²) / (x² + 7x + 12) ) / ( (2x - 4)/ (x + 3) )

Answer:

First thing, split the fraction multiplying by the reciprocal of the denominator:

(4 - x²) / (x² + 7x + 12) * (x + 3) / (2x - 4)

After this, you can factor the quadratics and then cross out common factors.

((2 - x)(2 + x)) / ((x + 3)(x + 4)) * (x + 3) / (2(x - 2))

((2 - x)(2 + x)) / ((x + 3)(x + 4)) * (x + 3) / (2(x - 2))

Note that (2 - x) is equal to (-1)(x - 2):

((-1)(x - 2)(2 + x)) / (x + 4)) * 1 / (2(x - 2))

((-1)(2 + x)) / (x + 4)) * 1 / (2)

(-2 - x) / (2(x + 4))

37. 7 During a particular month, a local company surveyed all its employees to determine their travel times to work, in minutes. The data for all 15 employees are shown below.

25 55 40 65 29
45 59 35 25 37
52 30 8 40 55 Determine the number of employees whose travel time is within one standard deviation of the mean.

Answer:

Put the data in the calculator and perform One Variable Statistics to find the mean and the standard deviation.

The mean is 40 and the standard deviation is 14.8. This means that we want the number of employees whose travel to are in the range of (40 - 14.9) to (40 + 14.9), or 25.1 - 54.9.

Make sure you list the standard deviation and the range. If you make a mistake, you can still get partial credit for a consistent answer.

There are 8 employees in that range: 40, 29, 45, 35, 37, 52, 30, 40.

38. The measures of the angles between the resultant and two applied forces are 60° and 45°, and the magnitude of the resultant is 27 pounds. Find, to the nearest pound, the magnitude of each applied force.

Answer:

If you create a parallelogram with a diagonal of 27, then the two forces can be drawn as the sides of the parallelogram. The diagonal forms 60 and 45 degree angles. You now have two triangles. The third angle is 180 - 60 - 45 = 75 degrees. Now you can use the Law of Sines to find the length of the sides:

Sin 60 / a = Sin 75 / 27
a = 27 sin 60 / sin 75 = 24.2 = 24 pounts.

Sin 45 / b = Sin 75 / 27
b = 27 sin 45 / sin 75 = 19.7 = 20 pounts.

End of Part III

More to come. Comments and questions welcome.

More Regents problems.

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