Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2014)

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 2014

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


36. Solve algebraically for all exact values of x in the interval 0 < x < 2π:

2 sin²x + 5 sin x = 3

Answer:

Start by setting it up like any other quadratic equal to 0.

2 sin²x + 5 sin x = 3
2 sin²x + 5 sin x - 3 = 0
(2 sin x - 1)(sin x + 3) = 0
2 sinx x - 1 = 0 or sin x + 3 = 0
2 sin x = 1 or sin x = -3 (discard)
sin x = 1/2

In the interval 0 < x < 2π, sin x = 1/2 at π/6 and 5π/6.

If you showed the above work, but answered 30 and 150, you would've lost a point.

37. Because Sam’s backyard gets very little sunlight, the probability that a geranium planted there will flower is 0.28. Sam planted five geraniums. Determine the probability, to the nearest thousandth, that at least four geraniums will flower.

Answer:

The probabilty that at least four will flower is the probabilty that four will flower plus the probability that five will flower. Or the probability that only 1 or none will not flower.

P(exactly 4 will flower) = (₅C₄)(0.28)⁴(0.72) = 0.02212...
P(5 will flower) = (0.28)⁵ = 0.00172...
P(4 or 5) = 0.02384 = 0.024

38. Two sides of a parallelogram measure 27 cm and 32 cm. The included angle measures 48°. Find the length of the longer diagonal of the parallelogram, to the nearest centimeter.

Answer:

If the included angle is 48 degrees, then the supplementary angle in the parallelogram will be 180 - 48 = 132 degrees. The diagonal opposite this angle will be the longer of the two.

Use the Law of Cosines (from the back of the booklet): a² = b² + c² - 2bc cos A, where a will be the diagonal, b and c are the given sides, and A is the included angle, in this case 132.

a² = b² + c² - 2bc cos A
a² = (27)² + (32)² - 2(27)(32)cos(132)
a² = 2909.257
a = 53.937... = 54 cm.

End of Part III.

More to come. Comments and questions welcome.

More Regents problems.

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