Algebra Problems of the Day (Integrated Algebra Regents, August 2012)

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

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August 2012

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


37. A company is running a contest and offering a first, second, and third prize. First prize is a choice of a car or $15,000 cash. Second prize is a choice of a motorbike, a trip to New York City, or $2,000 cash. Third prize is a choice of a television or $500 cash.

If each prize is equally likely to be selected, list the sample space or draw a tree diagram of all possible different outcomes of first, second, and third prizes.

Determine the number of ways that all three prizes selected could be cash.

Determine the number of ways that none of the three prizes selected could be cash.

Answer:

The sample space would be (you can abbreviate, but be clear about it):
Car, motorbike, TV
Car, motorbike, $500
Car, NYC, TV
Car, NYC, $500
Car, $2000, TV
Car, $2000, $500
$15K, motorbike, TV
$15K, motorbike, $500
$15K, NYC, TV
$15K, NYC, $500
$15K, $2000, TV
$15K, $2000, $500

The tree diagram would look like this:

The tree diagram or sample space is considered the WORK for the next two questions. If you answer the questions based on what you drew, you can get credit even if you made a mistake in your diagram, as long as it is consistent.

If you did NOT create a tree diagram or sample space, then work is expected to be shown on the bottom two questions, such as using the Counting Principle.

There is only ONE way that all three prizes will be cash. ($15,000, $2,000, $500)

There are TWO ways that none of the three prizes will be cash. (Car, bike, TV, and Car, NYC, TV)

38. In right triangle ABC shown below, AC = 29 inches, AB = 17 inches, and m∠ABC = 90. Find the number of degrees in the measure of angle BAC, to the nearest degree.

Find the length of BC to the nearest inch.

Answer:

You are given that it is a right triangle, along with the length of the hypotenuse and one of the sides. You want to find the size of the angle ADJACENT to the side you know. That means using Cos.

Cos x = 17/29
x = cos^-1(17/29) = 54.11...

The measure of ∠BAC is 54 degrees.

To find the length of BC, you have a choice of using the Sine ratio, or Pythagorean Theorem.

17² + BC² = 29²
289 + BC² = 841
BC² = 552
BC = 23.49...
BC = 23 inches

Or...

sin 54.11 = BC/29
BC = 29 * sin 54.11 = 23.49...
BC = 23 inches

39.On the set of axes below, graph the following system of inequalities.

y + x ≥ 3
5x - 2y > 10

State the coordinates of one point that satisfies y + x ≥ 3, but does not satisfy 5x - 2y > 10.

Answer:

Rewrite the inequalities in slope-intercept form, so you can use your calculator, or just plot from the y-intercept and the slope.

y + x ≥ 3
y ≥ -x + 3

5x - 2y > 10
- 2y > -5x + 10
y < 5/2 x - 5

Remember to shade on the correct sides. Remember that the second inequality is a broken or dashed line.

You may choose any point that is in the top quarter of the graph, the top of the "X", for the final answer. You may choose a point on the top part of the red line, the top part of the blue line, or any point in between them. For example, (2, 2).

End of Exam.

More to come. Comments and questions welcome.

More Regents problems.