Integrated Algebra Regents Exam (August 2013) Discussion, Part II, III and IV

EDIT: Welcome to my website. If you like the information in this blog entry or find it useful, please, feel free to leave a comment. Thank you for visiting.

Part II

31. A jogger ran at a rate of 5.4 miles per hour. Find the jogger's exact rate, in feet per minute. Note: “1 mile = 5280 feet” was given.

To convert from miles per hour to feet per minute, you need to multiply by 5280 feet/minute and 1 hour/60 min. Why is the 60 in the bottom? Look at this

The hour units cancel as do the mile units, leaving only feet per minute. Evaluate the expression and you get 475.2 feet per minute.

Partial credit for doing one step, or miscalculating or for rounding to the nearest foot.

32. Simplify twice the square root of 108, which I’ll write as 2(108)^.5
A reminder: “^.5” means “raised to the one-half power”, which means to take the square root. You can use this form in spreadsheets.

The largest perfect square that is a factor of 108 is 36. So you have (2)(36*3)^.5, which is (2)(6)(3)^.5, which is 12(3)^.5, or 12 times the square root of 3.

Partial credit if you didn't simplify completely, or if you gave the decimal equivalent (which in my opinion shouldn't be worth anything, but, as it has been noted, I don't write the exams).

33. Adrianne invested $2000 in an account at a 3.5% interest rate compounded annually. She made no deposits or withdrawals on the account for 4 years. Determine, to the nearest dollar, the balance in the account after the 4 years.

Compounded interest. (2000)(1.035)⁴ = 2295.046, which is $2295 to the nearest dollar.

I don't know if there's credit for calculating simple interest. A point is lost for not rounding correctly, which on a two-point question hurts a lot!

Part III

34. Miller's Department Store is having a sale with a 25% discount on mattresses. If the sales tax rate is 8%, how much change will Frank receive from $800 if he purchases a mattress regularly priced at $895 during this sale?

In one line (but don't do this in one line), it's:

800 - (895 * .75)(1.08) = 800 - 724.95 = 75.05

Find 75% of $895 (or find 25% of it and then subtract that amount from 895) and multiply by 1.08 (or find 8% of the subtotal and then ADD that to the subtotal) to get the total amount spent. Subtract that from $800 to find out how much change he'll get.

Partial credit if you stopped along the way or made a calculation error.

35. The difference between two numbers is 28. The larger number is 8 less than twice the smaller number. Find both numbers. Let x be the smaller number and y be the bigger number. Then the two equations that you need are

y - x = 28
y = 2x - 8
Now, this is a good case for substitution (unlike question 15).
2x - 8 - x = 28
x - 8 = 28
x = 36 (the smaller number)
so y = 36 + 28 = 64 or y = 2(36) - 8 = 72 - 8 = 64. (check!)

Partial credit for only finding x and forgetting about y. Checking isn't required, but it's a good idea. If you got the wrong x, but evaluated y using that x, that's consistency, so it's partial credit.

36. Janis measures the dimensions of the floor in her rectangular classroom for a rug. Her measurements are 10.50 by 12.25 feet. The actual measurements of the floor are 10.75 feet by 12.50 feet. Determine the relative error in calculating the area, to the nearest thousandth.

Hey, tutoring class! We did a similar problem, only with Volume. This one is an easier example of this type of question compared to what they usually ask. Make sure you round correctly.

Find the two areas. Then calculate (the difference of the areas)/(the actual area).

(134.375 - 128.625)
134.375
= 0.042790..., which is 0.043 to the thousandth place.

The two areas together are worth a single point. The calculations give you the next point. The final answer, rounded correctly, give you the third point.

37. A graph. Absolutely no reason that you should lose any points if you plot the graph correctly, label the lines and state the solutions (the points of intersection).

And if you realize that y + 1 = -2x is the same as y = -2x - 1, you can put both equations into your graphing calculator, graph it and look at the Table of Values. All you would need to do is copy the points onto your paper and draw the lines. They meet at (-2, 3), which must be labeled.

If you graphed a parabola with an error or an incorrect line (ex, y = -2x + 1), the lines should still intersect and you can state (approximately, if necessary) the point or points of intersection. If your graph is so far off that there are no points of intersection, then you would have written "No solution.", although that should have tipped you off that there's a mistake.

38. It's easier to use a picture than to explain:

No points if you canceled out all the x²'s.

39. A bottle contains 12 red marbles and 8 blue marbles. A marble is chosen at random and not replaced. Then, a second marble is chosen at random.
a. Determine the probability that the two marbles are not the same color.

You need to find P(red then blue) and P(blue then red), and add them. (They'll be the same.)
so (12/20 * 8/19) + (8/20 * 12/19) = 192/380

b. Determine the probability that at least one of the marbles is red. You can find P(Red then blue) and P(blue then red) -- which we just did -- and P(two reds) and add them.
So (192/380) + (12/20 * 11/19) = (192/380) + (132/380) = 324/380

Alternatively, you could find the probability that both are blue and then subtract that from 1:
1 - (8/20 * 7/19) = 1 - (56/380) = (380/380) - (56/380) = 324/380.