Algebra Problems of the Day (Integrated Algebra Regents, June 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

June 2012

Part IV: Each correct answer will receive 4 credits. Partial credit is possible.


37. Mike buys his ice cream packed in a rectangular prism-shaped carton, while Carol buys hers in a cylindrical-shaped carton. The dimensions of the prism are 5 inches by 3.5 inches by 7 inches. The cylinder has a diameter of 5 inches and a height of 7 inches.

Which container holds more ice cream? Justify your answer.

Determine, to the nearest tenth of a cubic inch, how much more ice cream the larger container holds.

Answer:

You need to find the Volume of both of these containers and state which one has more.

Volume of a rectangular prism is V = L W H = (5)(3.5)(7) = 122.5 cubic inches

Volume of a cylinder is V = pi R² H = (3.141592)((1/2)(5))²(7) = 137.4

The cylinder holds more.

Note that if you made a computational error, like forgetting to take half of the diameter, you would have lost a point even though the cylinder would still be more.

137.4 - 122.5 = 14.9

If you hadn't been required to find the difference of the Volumes, you could have just compared the two:

Is the Prism > the Cylinder ?

(5)(3.5)(7) > (3.141592)((1/2)(5))²(7) ?

(5)(3.5) > (3.141592)(2.5)² ?

(5)(3.5) > (3.141592)(6.25) ?

(3.5) > (3.141592)(1.25) ?

(3.5) > 3.92699 Not true.

The cylinder holds more.

38. Solve algebraically for x: 3(x + 1) - 5x = 12 - (6x - 7)

Answer:

Unecessarily complicated, but it's a Part IV question. Surprisingly, it's not even quadratic!

3(x + 1) - 5x = 12 - (6x - 7)
3x + 3 - 5x = 12 - 6x + 7
-2x + 3 = 19 - 6x
4x + 3 = 19
4x = 16
x = 4

You're not required to check your work, but you can. You probably should if it seemed too easy, or was too difficult.

Also, notice that they don't tell you to round your answer, so if you got an infinite decimal, you probably made a mistake.

Check:

3(4 + 1) - 5(4) = 12 - (6(4) - 7)
3(5) - 20 = 12 - (24 - 7)
15 - 20 = 12 - 17
-5 = -5
Check!

39. A large company must chose between two types of passwords to log on to a computer. The first type is a four-letter password using any of the 26 letters of the alphabet, without repetition of letters. The second type is a six-digit password using the digits 0 through 9, with repetition of digits allowed.

Determine the number of possible four-letter passwords.

Determine the number of possible six-digit passwords.

The company has 500,000 employees and needs a different password for each employee. State which type of password the company should choose. Explain your answer.

Answer:

Use the Counting Principle. Without repetition means that the number of possibilities is going to decrease with each choice. With repetition means that the number will not change.

Four-letter PWs: 26 * 25 * 24 * 23 = 358,800

Six-digit PWs: 10 * 10 * 10 * 10 * 10 * 10 = 10⁶ = 1,000,000

The company should use the six-digit PWs because there are not 500,000 four-letter PWs available to use.

End of Part IV.

End of Exam.

More to come. Comments and questions welcome.

More Regents problems.