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 II: Each correct answer will receive 2 credits. Partial credit is available.


28. Show that sec θ sin θ cot θ = 1 is an identity.

Answer:

Remember that sec θ = 1 / cos θ and cot θ = 1 / tan θ = cos θ / sin θ.

So

sec θ sin θ cot θ = 1
1 / cos θ * sin θ * cos θ / sin θ = 1
(sin θ * cos θ) / (cos θ * sin θ) = 1
1 / 1 = 1

29. Find, to the nearest tenth of a square foot, the area of a rhombus that has a side of 6 feet and an angle of 50°.

Answer:

A rhombus can be divided into four congruent right triangles. The side of the rhombus will be the hypotenuse of the triangle. The given angle will be bisected by the diagonal, meaning that the angles in the triangle will be 25, 90 and 65 degrees. You can use sine and cosine to find the lengths of the base and the height to find the Area of one of the triangles and multiply that by four.

The quicker way is that the area of a rhombus can be found with the formula A = b² * Sin(a)

A = (6)² * sin(50) = 27.57... = 27.6 to the nearest tenth of a square foot.

The LONG way:

6 * Sin (25) = 6(.422...) = 2.535...
6 * cos (25) = 6(.906...) = 5.437...
A = 4 * (1/2) * (2.535) * (5.437) = 27.56..., which rounds to 27.6.
Note the extra accuracy used in the intermediate steps to prevent rounding errors.

30. The following is a list of the individual points scored by all twelve members of the Webster High School basketball team at a recent game:

2 2 3 4 6 7 9 10 10 11 12 14
Find the interquartile range for this set of data

Answer:

The Interquartile Range (IQR) is the difference between the Upper Quartile (Q3) and the Lower Quartile (Q1). The data are in order, so finding these is easy. Mark up the page to show your work.

There are twelve pieces of data, so the median is the average of the 6th and the 7th. Q1 is the middle of the lower six values, which is the average of the 3rd and 4th, or 3.5. Q3 is the middle of the upper six values, which is the average of the ninth and tenth, or 10.5

The IQR = 10.5 - 3.5 = 7.

31. Determine algebraically the x-coordinate of all points where the graphs of xy = 10 and y = x + 3 intersect

Answer:

Use substitution to solve the system of equations.

xy = 10 and y = x + 3
x(x + 3) = 10
x² + 3x = 10
x² + 3x - 10 = 0
(x + 5)(x - 2) = 0
x + 5 = 0 or x - 2 = 0
x = -5 or x = 2

More to come. Comments and questions welcome.

More Regents problems.

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