Algebra Problems of the Day (Integrated Algebra Regents, August 2010)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones. The Integrated Algebra Regents covered most of the same material as the current Algebra Regents, with a few differences.

More Regents problems.

Integrated Algebra Regents, August 2010

Part I: Each correct answer will receive 2 credits.


16. Which point lies on the line whose equation is 2x − 3y = 9?

1) (-1,-3)
2) (-1,3)
3) (0,3)
4) (0,-3)

Answer: 4) (0,-3)

Before you start plugging them all in, notice that the y-coordinates are 3 and -3, and -3(-3) = +9. This works if that 2x = 0 and therefore x = 0.

So Choice (4) is the solution.

You can try the other points and you'll get the following:

2(-1) - 3(-3) = 7
2(-1) - 3(3) = -11
2(0) - 3(3) = -9

17. Which phrase best describes the relationship between the number of miles driven and the amount of gasoline used?

1) causal, but not correlated
2) correlated, but not causal
3) both correlated and causal
4) neither correlated nor causal

Answer: 3) both correlated and causal

Reminder: the word is "causal", as in "cause and effect", not "casual" meaning relaxed or not permanent.

The more miles you drive, the more gas you will use. There is a correlation between the two variables. Driving more will cause more gas to be used, so it's causal.

Ice cream sales and bathing suit sales, for example, are correlated, but one doesn't cause the other. Both increase when the temperature gets warmer.

18. The height, y, of a ball tossed into the air can be represented by the equation y = −x² + 10x + 3, where x is the elapsed time. What is the equation of the axis of symmetry of this parabola?
1) y = 5
2) y = -5
3) x = 5
4) x = -5

Answer: 3) x = 5

The axis of symmetry is a vertical line with an equation x = a. Eliminate Choices (1) and (2) which are horizontal lines.

You can graph the equation in your calculator to see where the vertex is. Or you can use the formula x = -b / (2a).

x = -(10) / (2(-1)) = -10/-2 = 5, which is Choice (3).

If yo forget the formula for Axis of Symmetry, it is the same of that quadratic formula when the square root expression is 0.

19. In the diagram below, MATH is a rectangle, GB = 4.6, MH = 6, and HT = 15.

What is the area of polygon MBATH?

1) 34.5
2) 55.5
3) 90.0
4) 124.5

Answer: 2) 55.5

There is no ONE formula to hande shapes like this one. You need to break it down into parts.

There is a rectangle and there is a triangle. If you remove the triangle from the rectangle, you get the polygon MBATH, which is a concave pentagon.

Remove means subtract. Subtract the area of the triangle from the area of the rectangle.

Area of MATH = L*W = (6)(15) = 90. Area of MBA = 1/2 b h = .5(4.6)(15) = 34.5

90 - 34.5 = 55.5, which is Choice (2).

Notice that Choice (1) is the area of the triangle only and that Choice (3) is the area of the rectangle only. Choice (4) is the sum of the two areas, instead of the difference.

20. This year, John played in 10 baseball games. In these games he had hit the ball 2, 3, 0, 1, 3, 2, 4, 0, 2, and 3 times. In the first 10 games he plays next year, John wants to increase his average (mean) hits per game by 0.5. What is the total number of hits John needs over the first 10 games next year to achieve his goal?

1) 5
2) 2
3) 20
4) 25

Answer: 4) 25

If John wants to increase his average by .5 hits per game over 10 games, that is 5 additional hits in the first 10 games, so Choices (1) and (2) are obviously incorrect. Eliminate them.

If he hit 2 + 3 + 0 + 1 + 3 + 2 + 4 + 0 + 2 + 3 times, then he had 20 hits. Five more is 25, which is Choice (4).

If all you did was figure out that he had 20 hits this year, then the only possible answer to increase his average is 25.

More to come. Comments and questions welcome.

More Regents problems.

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