Monday, October 25, 2021

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 I: Each correct answer will receive 2 credits.

25. As shown in the table below, a person’s target heart rate during exercise changes as the person gets older.

Which value represents the linear correlation coefficient, rounded to the nearest thousandth, between a person’s age, in years, and that person’s target heart rate, in beats per minute?

1) -0.999
2) -0.664
3) 0.998
4) 1.503

Answer: 1) -0.999

A quick glance at the table will tell you that there is a negative correlation between the two columns, age and heart rate. That maeans that the coefficient of correlation will be negative. So you can eliminate choices (3) and (4).

Furthermore, since the coefficient cannot be greater than 1 (or less than -1), Choice (4) was never a possible answer.

To find the exact value of r, the coefficient of correlation, you would need to make a likst in your graphing calculator, and then run a linear regression. HOWEVER, that isn't necessary for this question.

Choice (1) says -0.999, which is just short of -1, which would be a straight line. A straight line has a constant rate of change. If the rate of change isn't constant (or incredibly close to constant), then r cannot be -1, or -0.999.

The Age value is going up by 5, and the Heart Rate column is always going down by 3 or 4. THis is very close to a straight line. So the answer is Choice (1).

Running a linear regression will lead you to the same outcome. (Remember to have DIAGNOSTICS ON on your calculator.)

26. In triangle MNP, m = 6 and n = 10. Two distinct triangles can be constructed if the measure of angle M is

1) 35
2) 40
3) 45
4) 50

Answer: 1) 35

Common sense (or maybe Number Sense?) may tell you to pick the smallest number. Or it might tell you that it has to be either the largest or the smallest number.

You can set up an equation with the Law of Sines:

Sin M / 6 = Sin N / 10
Then Sin N = 10 / 6 * Sin M
And N = Sin-1 (10/6 Sin M)

For this equation to work, the value of 10/6 sin M must be less than 1.000. Which means that Sin M must be less than .600.

Sin 35 = 0.57. Sin 40 = 0.64. So only the 35 degree angle will work.

When M = 35, then Sin N = 0.956, which gives N the possible values of 73 or 90 + (90 - 73) = 107.

27. If order does not matter, which selection of students would produce the most possible committees?

1) 5 out of 15
2) 5 out of 25
3) 20 out of 25
4) 15 out of 25

Answer: 4) 15 out of 25

If the order doesn't matter, then this is a Combination question.

Here is the Common Sense answer: 5 out of 25 will produce more possibilities that 5 out of 15. ANd 20 out of 25 is the SAME as 5 out of 25 because of symmetry. (The number of ways that you can Choose 5 of 25 is the same as the number of ways that you DO NOT CHOOSE 20 of 25.) You have no eliminated three of the four choices. The answer must be 15 out of 25.

Now consider Pascal's Triangle. If element in each row of Pascal's Triangle is a valuation of an nCr expression. If you look at an image of Pascal's Triangle, you will see that the numbers are larger as you get closer to the center.

And 15 out of 25 is closer to the center (half of 25 is 12.5) then either 5 or 20.

Finally, if you put the four expressions in your calculator, 25C15 gives you a number greater than 3 million.

The other choices give you 3003, 53130 and 53130, respectively.

End of Part I

More to come. Comments and questions welcome.

More Regents problems.

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