August 2015 Algebra 1 (Common Core) Regents: Parts 3 and 4

Here are the questions, with answers and explanations, for the New York State Algebra 1 (Common Core) Regents exam, Parts 3 and 4. There were 4 questions in Part 3, each worth 4 credits. There is one question in Part 4, worth 6 credits. Partial credit may be earned for correct work on a problem without a solution, or for a problem with a solution that contains computational or conceptual errors. All work must be shown. In general, a correct answer without any work is worth 1 credit, unless that answer is given as a choice and an explanation is required.

Link to Part 1
Link to Part 2

Part 3

33. Graph f(x)= x² and g(x) = 2^x for x ≥ 0 on the set of axes below.
State which function, f(x) or g(x), has a greater value when x = 20. Justify your reasoning.

They did not specify a scale, but it you used each box is one, it will be hard to read. Make sure you label it clearly. You might want to include your Table of Values from your calculator. If you use a scale of 1, you will reach the top of the graph by x = 4.

f(20) = (20)² = 400; g(20) = (2)²⁰ = 1048576. g(20) > f(20).
You could also justify it from the graph if the graph clearly shows that g(x) is increasing more steeply than f(x) for x > 4.
An alternate graph, which is more obvious:

34. Solve for x algebraically: 7x - 3(4x - 8) ≤ 6x + 12 - 9x
If x is a number in the interval [4,8], state all integers that satisfy the given inequality. Explain how you determined these values.

7x - 3(4x - 8) ≤ 6x + 12 - 9x
7x - 12x + 24 ≤ -3x + 12
-5x + 24 ≤ -3x + 12
-2x ≤ - 12
x > 6
The integers 6, 7, 8 satisfy the condition because there are greater than or equal to 6 and in the interval [4, 8].

Seriously, I'm not sure what kind of explanation is needed if you solved the inequality, which explains the answer. They are in the solution set of the equation.

35. The volume of a large can of tuna fish can be calculated using the formula V = πr²h. Write an equation to find the radius, r, in terms of V and h.
Determine the diameter, to the nearest inch, of a large can of tuna fish that has a volume of 66 cubic inches and a height of 3.3 inches.

Look at the work in the image below: r = the square root of (V divided by (i times h))

Plug is V = 66 and h = 3.3 and solve. Volume divided by (pi times height) is approximately 6.366. The square root of that is approximated 2.52, so to the nearest inch, the diameter is 2.5 * 2 = 5 inches.

Don't forget to double the radius. I'm shocked that this was to the nearest inch when the height was given to the nearest tenth of an inch.

36. The table below shows the attendance at a museum in select years from 2007 to 2013.

State the linear regression equation represented by the data table when x = 0 is used to represent the year 2007 and y is used to represent the attendance. Round all values to the nearest hundredth.
State the correlation coefficient to the nearest hundredth and determine whether the data suggest a strong or weak association.

Subtract 2007 from all the years to get 0, 1, 2, 4, and 6. Put those numbers and the second row of data into two lists on the calculator. Perform a linear regression. You should get y = 0.16x + 8.27. The correlation coefficient is 0.97, with is a strong association. (Anything close to 1 or close to -1 is a strong association.)

Part 4

37. 7 A rectangular picture measures 6 inches by 8 inches. Simon wants to build a wooden frame for the picture so that the framed picture takes up a maximum area of 100 square inches on his wall. The pieces of wood that he uses to build the frame all have the same width.

Write an equation or inequality that could be used to determine the maximum width of the pieces of wood for the frame Simon could create.

Explain how your equation or inequality models the situation.

Solve the equation or inequality to determine the maximum width of the pieces of wood used for the frame to the nearest tenth of an inch.

You have to love that they won't tell you whether to use an equation or an inequality. (Okay, you don't really have to love it at all.)

The wood has width x, and is added to both sides of the picture, plus the top and bottom. Area = L X W.
So (2x + 6)(2x + 8) < 100
4x² + 28x + 48 < 100

Explain: the wood adds 2x to the length of the picture and the width of the picture. The area cannot be more than 100 square inches.

Solve: first, divide by 4: x² + 7x + 12 < 25
subtract 25 from both sides: x² + 7x - 13 < 0
From here you need to use the quadratic formula or complete the square. Let's complete the square: x² + 7x + 49/4 - 13 < 49/4

x² + 7x + 49/4 < 12.25 + 13
(x + 7/2)² < 25.25
x + 7/2 < 5.024 or x + 7/2 > -5.024. Discard the negative answer.
x < 1.524 The width cannot be more than 1.5 inches.

Okay, I probably should've gone with the quadratic formula. Probably would've been more straightforward, and you wouldn't have had all those halves and quarters.

End of exam.