Friday, May 02, 2014

January 2014 Algebra Regents, Thread 2

It's Day 7, if anyone is keeping track

Continuing with the multiple-choice questions:

January 2014 Algebra Regents

11. The graph below illustrates the number of acres used for farming in Smalltown, New York, over several years.

Using a line of best fit, approximately how many acres will be used for farming in the 5th year?

The trend is downward. It will be below 300, but not so low as to hit zero. The only logical choice is 200.

12. When 16x3 - 12x2 + 4x is divided by 4x, the quotient is

Divide each co-efficient by 4, subtract one from each exponent. The last term (4x) is divided by itself, leaving + 1. It's choice (4).

If you don't understand why it isn't choice (2), this comic may be relevant!

13. The width of a rectangle is 4 less than half the length. If ℓ represents the length, which equation could be used to find the width, w?

Translating the words into an equation W = (1/2) L - 4. Pretty straightforward.

14. Which data can be classified as quantitative?

Quantitative is a numeric value that count or add or average or compute. Something that isn't a description (In other words, not a zip code, phone number or shoe size).

Three choices are things. Only the sales tax rate is a measurable amount.

15. Two cubes with sides numbered 1 through 6 were rolled 20 times. Their sums are recorded in the table below.

What is the empirical probability of rolling a sum of 9?

Theoretically, the odds of a 9 are 4/36 = 1/9. However, that's not what they asked. An experiment was performed and we need to use the results of the experiment. (That's what "empirical" means.) Who knows? Maybe the dice are weighted slightly and, therefore, biased. Count the number of times 9 appears in the table. Then go back and do it again! Did you get the same number both times? If so, there's a good chance you didn't miss any. If you didn't, you might want to check a third time, just to be sure.

I counted five of them. Both times. Five out of twenty is 5/20 (which simplifies to 1/4, but they didn't simplify their answers).

16. What is the vertex of the graph of the equation y = 3x2 + 6x + 1?

You can answer this by entering it in your graphing calculator and taking a look. But if you don't want to, here's another way:

The x-co-ordinate of the vertex can be found with the formula x = (-b)/(2a). So x = (-6)/(2*3)= (-6)/(6) = -1. Notice that two choices have been eliminated. Now substitute -1 for x and evaluate for y. y = 3(-1)^2 + 6(-1) + 1 = 3(1) + (-6) + 1 = -2. The vertex is (-1, -2), which is choice (1).

17. The length and width of a rectangle are 48 inches and 40 inches. To the nearest inch, what is the length of its diagonal?

Use the Pythagorean Theorem and hit the square root button rather than simplify any radical numbers. c^2 = 48^2 + 40^2 = 2304 + 1600 = 3904. The square of 3904 is 62.48, which rounds down to 62. Choice (2).

NUMBER SENSE ALERT!! Common sense will tell you that the diagonal is longer than the two sides, so 27 is out. Common sense will also tell you that the third side of a triangle MUST be shorter than the sum of the other two sides. So 88 and 90 are out. There's only one possible choice.

18. Which graph represents the solution set of 2x - 5 < 3?

Add 5 to both sides (3 becomes 8) and then divide by 2 (8 becomes 4), so x < 4. That's a ray with an open circle at one end and pointing to the left. Choice (1).

19. Jonathan drove to the airport to pick up his friend. A rainstorm forced him to drive at an average speed of 45 mph, reaching the airport in 3 hours. He drove back home at an average speed of 55 mph. How long, to the nearest tenth of an hour, did the trip home take him?

If it took 3 hours going 45 mph to get there, then Distance = Rate X Time = 45 X 3 = 135. The airport is 135 miles away.
T = D / R = 135 / 55 = 2.454545...., which is approximately 2.5 hours, to the nearest tenth. Choice (2).

20. The expression (2n)/5 + (3n)/2 is equivalent to

Forgive me for trying to minimize the number of images that I need to upload. Those were two fractions, without common denominators.
Multiply the first fraction by 2/2 and the second by 5/5. The numerators become 4n and 15n, respectively. They total to 19n over 10. Choice (3).

Note: If you said (5n)/7, you have bigger problems than Algebra. No, seriously. I'm not trying to be mean, You need to review the basics before you can move forward, or you will forever be scratching your head at incorrect responses.

21. When x = 4, the value of 2x0 + x! is

FLASHBACK! JUNE 2013!!: "Mr. Burke, what does an exclamation point in a math problem mean?" -- A frustrated student who ran into me after the Regents exam.

Mini-rant: I hate when factorials (the exclamation point !) are added to problems just because they can be. It has a purpose, and this isn't it.

For those who, like my former student, don't recall ever seeing an exclamation point before, it's called a "factorial", and what it means is this: multiply the number by the number before it times the number before that all the way down to 1. So 5! = 5 X 4 X 3 X 2 X 1 = 120. There are important uses for this is Statistics and Probability. For example, the number of ways that you can arrange five pictures in a row on a shelf is 5!.

This problem, however, is just stupid. On the other hand, it's "theoretical". In theory, there's no reason why we can't answer it -- even if it doesn't make sense to us.

Order of Operations: Substitute (4) for x. Take (4) to the 0th power, which is 1. Multiply 1 by 2 and you get 2. The addition has to wait because the factorial takes precedence because it is a form of exponentiation. (For anyone wanting to argue that point, consider this alternative explanation 2*3! is NOT equal to 6!, so the multiplication comes second!)

4! = 4 X 3 X 2 X 1 = 24. 2 + 24 = 26. Choice (3)

22. Which graph represents the solution of 2y + 6 > 4x?

First of all, it's a broken line because it's "greater than" and not "greater than or equal to". Eliminate two choices. Check if the Origin (0, 0) is in the solution. Is 0 + 6 > 4(0)? Yes, it is! So you have to shade the side of the line that includes (0, 0). That's choice (3).

Notice that you didn't have to put the problem into slope-intercept form, or graph it in any way to get the answer. Number Sense pays off again!

23. Which graph represents the exponential decay of a radioactive element?

Radioactive decay drops fast and then slows down, getting closer to zero (the x-axis) without ever meeting or crossing it. That's Choice (4).

24. Which fraction represents (x2 - 25) / (x2 - x - 20)expressed in simplest form?

Factor! Factor! Factor!

If you cancel out the two x2, then get out of my classroom!!!!! No! You can't do that!

Factor the numerator into (x + 5)(x - 5). It's the Difference of Squares Rule. Factor the numerator into (x - 5)(x + 4).

Cancel out the two (x - 5) factors, and you are left with (x + 5) / (x + 4).

25. If abx - 5 = 0, what is x in terms of a and b?

Add 5 to both sides. abx = 5. Divide by the co-efficient of x, which is ab, and x = 5 / (ab).

Remember: Subtract the constant and divide! You'll find your x and you'll know why! or so the song goes.

26. Given: U = {x|0 < x < 10 and x is an integer}, S = {x|0 < x < 10 and x is an odd integer}
The complement of set S within the universal set U is

The complement of all the odd integers between 0 and 10, exclusive, within the Universal set of integers between 0 and 10, exclusive, would be the even integers between 0 and 10, exclusive. I say "exclusive" because of the "less than" signs, which indicate that 0 and 10 are NOT part of the solution. So it's {2, 4, 6, 8}, pick Choice (4), that's really great!

27. The roots of the equation 2x2 - 8x = 0 are

Divide by two to get: x2 - 4x = 0, which factors into (x)(x-4) = 0. So either x = 0 or x - 4 = 0. Therefore, x = 0 or x = 4. Choice (4).

28. Which equation illustrates the multiplicative inverse property?

Another definition question. You either know it or you don't. The inverse of a multiplication problem is its reciprocal. A number times its reciprocal equal 1. Choice (3).

29. What is the result when 4x2 - 17x + 36 is subtracted from 2x2 - 5x + 25?

First of all, subtracted from means taking the first from the second. As in "15 subtracted from 20" means "20 - 15". Don't do the problem backward!

Look at the co-efficients: 2 - 4 = -2; -5 - (-17) = +12. You know enough to get your answer, but check: 25 - 36 = -11. Choice (4)

30. Julie has three children whose ages are consecutive odd integers. If x represents the youngest child’s age, which expression represents the sum of her children’s ages?

Consecutive odd integers are x, x + 2, x + 4, etc. If there are three numbers, then their sum is x + x + 2 + x + 4, which is 3x + 6. Choice (4)

to be continued . . .

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