Monday, February 05, 2018

January 2018 Common Core Algebra I Regents, Parts 3 and 4

The following are some of the open-ended questions from the recent January 2018 New York State Common Core Algebra I Regents exam, parts 3 and 4
Part II can be found here.

January 2018 Algebra I, Part III

Each correct answer is worth up to 4 credits. Work must be shown.

33. Jim is a furniture salesman. His weekly pay is $300 plus 3.5% of his total sales for the week. Jim sells x dollars' worth of furniture during the week. Write a function, p(x), which can be used to determine his pay for the week.

Use this function to determine Jim's pay to the nearest cent for a week when his sales total is $8250.

Answer: p(x) = 300 + .035x
$300 is the fixed amount, .035 is 3.5% in decimal form, x is the variable.

To find his pay, substitute 8250 for x.
P(8250) = 300 + .035(8250)
P(8250) = $588.75

34. Omar has a piece of rope. He ties a knot in the rope and measures the new length of the rope. He then repeats this process several times. Some of the data collected are listed in the table below.

State, to the nearest tenth, the linear regression equation that approximates the length, y, of the rope after tying x knots.
Explain what the y-intercept means in the context of the problem.
Explain what the slope means in the context of the problem.

Answer: Put all of the values into L1 and L2 on your graphing calculator. Then perform a Linear regression to find the slope (a) and y-intercept (b)
This will give you y = -8.5x + 99.2

The y-intercept means that when there are no knots in the rope, the length of the rope will be 99.2 cm.

The slope of the equation means that each knot made will decrease the length of the rope by 8.5 cm.

35. The drama club is running a lemonade stand to raise money for its new production. A local grocery store donated cans of lemonade and bottles of water. Cans of lemonade sell for $2 each and bottles of water sell for $1.50 each. The club needs to raise at least $500 to cover the cost of renting costumes. The students can accept a maximum of 360 cans and bottles.

Write a system of inequalities that can be used to represent this situation.

The club sells 144 cans of lemonade. What is the least number of bottles of water that must be sold to cover the cost of renting costumes? Justify your answer.

Answer: Let L = the number of cans of lemonade and W = the number of bottles of water.
Note: Use a capital L or a script l -- don't let your L's look like ones!

2L + 1.5W > 500
L + W < 360

2L + 1.5W > 500
2(144) + 1.5W > 500
288 + 1.5W > 500
1.5W > 212
W > 141.33333

Round up to 142 bottles.
Do NOT round down. That won't be enough -- you will have less than $500.

36. A manager wanted to analyze the online shoe sales for his business. He collected data for the number of pairs of shoes sold each hour over a 14-hour time period. He created a graph to model the data, as shown below.

The manager believes the set of integers would be the most appropriate domain for this model. Explain why he is incorrect.

State the entire interval for which the number of pairs of shoes sold is increasing.

Determine the average rate of change between the sixth and fourteenth hours, and explain what it means in the context of the problem.

Answer: The manager is incorrect. (Don't forget to state this with your explanation.)
The most appropriate domain would be whole numbers numbers which are positive and zero. Integers would include negative numbers, but you cannot sell a negative number of shoes.

The number of pairs of shoes sold is increasing during the interval 0 < t < 6.
After 6 hours, fewer pairs are being sold each hour.

To find the average rate of change, use the slope formula for the sixth hour (6, 120) and the fourteenth hour (14, 0)

slope = (y2 - y1) / (x2 - x1) = (0 - 120) / (14 - 6) = (-120) / 8 = -15
In the context of the problem, this means that 15 fewer shoes were being sold each hour between hours 6 and 14.

EDIT: The scoring guidelines were updated after the exam to allow for "0 to 14" as an acceptable answer because of the imprecise wording of the problem.

January 2018 Algebra I, Part IV

This answer is worth up to 6 credits. Work must be shown.

37. At Bea's Pet Shop, the number of dogs, d, is initially five less than twice the number of cats, c. If she decides to add three more of each, the ratio of cats to dogs will be 3/4.
Write an equation or system of equations that can be used to find the number of cats and dogs Bea has in her pet shop.

Could Bea's Pet Shop initially have 15 cats and 20 dogs? Explain your reasoning.

Determine algebraically the number of cats and the number of dogs Bea initially had in her pet shop.

Note: I'd have to check the archives, but I seem to remember a very similar question years ago.

Answer: Let d = the number of dogs and c = the number of cats
The first equation can be translated directly from the first sentence:

d = 2c - 5
Remember that "five less than" means "- 5".
The next sentence gives us two ratios, so the second equation is a proportion:
(c + 3)/(d + 3) = 3/4

Second part: Plug in the values for 15 cats and 20 dogs.
Check (20) = 2(15) - 5?
20 = 30 - 5 = 25, Incorrect.
No, 15 cats and 20 dogs cannot be the initial number of pets because the first equation would not be true.

Third part: You need to solve by substitution and then cross-multiplying the proportion.
Substitute (2c - 5) for d in the proportion.

Bea initially had 9 cats and 13 dogs.

End of exam.

How did you do?

Questions, comments and corrections are welcome.


George said...

Regarding the shoe salesman, I disagree with your answer for the interval over which "pairs of shoes sold is increasing." The answer is 0<6<14 . I'm sure you'd agree that the number of shoes sold at hour 12 is greater than the number of shoes sold at hour 11. Sure, the rate has decreased, but the number of shoes sold is increasing throughout the domain of this graph.

The question writer probably meant to say "at what interval is the RATE of shoe sales increasing?" (but didn't)

I also disagree with your assertion "time can't be negative." Ever watch a rocket launch? T-minus 10 etc. This graph, at time zero, refers to the previous hour according to the label of the y-axis.

(x, why?) said...

Good morning.

Yes, the scoring guidelines were updated after the exam to allow for 0 to 14 (or 0 < t < 14) as an acceptable answer as well as 0 to 6 (or 0 < t < 6) because of the imprecise wording of the question. However, 0 < 6 < 14 would likely been scored as imprecise, as there's no need to specify a middle number, just the beginning and end. (Seriously, I've seen things like this marked wrong because of the guidelines, and arguing otherwise wouldn't have made a difference.)

As to negative time, the manager was not preparing to launch the shoes into orbit. With regard to this graph, negative numbers -- hours before the store opened? -- do not make sense.

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