## Thursday, July 25, 2024

### Algebra Problems of the Day (Algebra Regents, June 2024 Part I)

This exam was adminstered in June 2024.

More Regents problems.

### Part I

Each correct answer will receive 2 credits. No partial credit.

1. A ball was launched into the air, and its height above the ground was recorded each second, as shown in the table below.

Based on these data, which statement is a valid conclusion?

(1) The ball lands on the ground at 4 seconds.
(2) The ball reaches a maximum height of 11 feet.
(3) The ball was launched from a height of 0 feet.
(4) The ball reaches its maximum height at 2 seconds.

Answer: (4) The ball reaches its maximum height at 2 seconds.

The ball starts at 11 feet from the ground, which is at 0 feet. Read the table to find the correct answer.

Choice (1) says the ball is on the ground at 4 seconds. It is not. It is at 11 feet. Eliminate Choice (1).

Choice (2) says the ball is reaches a maximum of 11 feet. No, it starts at 11 feet and goes up to 75 feet. Eliminate Choice (2).

Choice (3) says the ball was launched from 0 feet. No, it starts at 11 feet at 0 seconds. Eliminate Choice (3).

Choice (4) says the ball reaches its maximum height at 75 feet. The point (2,75) is the highest in the table. Also, the numbers on either side are symmetrical, so you can be sure that (2,75) is the vertex of the parabola that these points would graph. This is the correct answer.

2. A tour bus can seat, at most, 48 passengers. An adult ticket costs \$18 and a child ticket costs \$12. The bus company must collect at least \$650 to make a profit. If a represents the number of adult tickets sold and c represents the number of child tickets sold, which system of inequalities models this situation if they make a profit?

(1) a + c < 48 ; 18a + 12c > 650
(2) a + c < 48 ; 18a + 12c > 650
(3) a + c < 48 ; 18a + 12c < 650
(4) a + c < 48 ; 18a + 12c < 650

Answer: (2) a + c < 48 ; 18a + 12c > 650

At most means "is less than or equal to". At least means "is greater than or equal to".

Immediately eliminate Choice (1) and (3) which don't have the "equal to" portion.

Choice (2) has the correct symbols that match the language in the question.

3. Which equation is always true?

(1) x2 • x3 = x5
(2) 3x • 32 = 92x
(3) -z2 = z2
(4) 7a • 7b = 7ab

Answer: (1) x2 • x3 = x5

When you multiply two terms that have the same base, you keep the base and add the exponents.

In Choice (1), x2 • x3 = x x • x x x = x5. This is the correct answer.

In Choice (2), 3x • 32 = 3(x+2), not 92x. Eliminate Choice (2).

In Choice (3), -z2 = z2 is not true for any value other that z = 0 because of the Order of Operations. What is true is (-z)2 = z2. Eliminate Choice (3).

In Choice (4), 7a • 7b = 7a+b , not 7ab. Eliminate Choice (4).

4. The expression -2(x2 - 2x + 1) + (3x2 + 3x - 5) is equivalent to

(1) x2 + x - 4
(2) x2 - x - 7
(3) x2 + 7x - 4
(4) x2 + 7x - 7

Answer: (4) x2 + 7x - 7

Use the distributive property and then combine like terms.

-2(x2 - 2x + 1) + (3x2 + 3x - 5)
-2x2 + 4x - 2 + 3x2 + 3x - 5
x2 + 7x - 7

5. Which sum is irrational?

(1) -2√(12) + √(100)
(2) -√(4) + 1/3 √(900)
(3) 1/2 √(25) + √(64)
(4) √(49) + 3√(121)

Answer: (1) -2√(12) + √(100)

You could put each of these into your calculator and see if you get an infinite, non-repeating decimal as answer. If the numbers under the radicals are perfect squares, then the numbers are rational numbers. The sum of two rational numbers is always rational. The sum of a rational and an irrational number is also irrational. (The sum of two irrational numbers is usually, but not always, irrational.)

Looking at the choices above, 12 is not a perfect square, but 100, 4, 900, 25, 64, 49, and 121 are perfect squares.

Choice (1) is the only choice that adds an irrational number to a rational number and is therefore irrational. This is the correct answer.

Choice (2) equals -2 + 10 = 8.

Choice (3) equals 2.5 + 8 = 10.5

Choice (4) equals 7 + 33 = 40.

6. The solution to (4(x - 5)) / 3 + 2 = 14 is

(1) 15
(2) 14
(3) 6
(4) 4

You can substitute each answer in your calcluator if you aren't sure how to solve this.

Use inverse operations to isolate the variable and solve for x:

(4(x - 5)) / 3 + 2 = 14
(4(x - 5)) / 3 = 12
4(x - 5) = 36
x - 5 = 9
x = 14

Check: (4(14 - 5)) / 3 + 2 = (4(9)) / 3 + 2 = (36) / 3 + 2 = 12 + 2 = 14. Check!

The correct answer Choice (2).

7. On an island, a rare breed of rabbit doubled its population each month for two years. Which type of function best models the increase in population at the end of two years?

(1) linear growth
(2) linear decay
(3) exponential growth
(4) exponential decay

Answer: (3) exponential growth

Continously doubling is growth not decay. The rate of change is not constant, so it is not linear.

This is an example of exponential growth. This is a definition question. There is nothing to solve. You either understand the concept or you do not.

8. What is the degree of the polynomial 2x - x2 + 4x3?

(1) 1
(2) 2
(3) 3
(4) 4

This is another concept/definition question. The degree of a polynomial is the value of the highest exponent. This is the term that would go first when written in standard form. The correct answer is 3, which is Choice (3).

Choice (1), 1, is the value of the exponent of the first term that is written, but it isn't the highest exponent. Eliminate Choice (1).

Choice (2), 2, is the coefficient of the first term that is written, not the exponent. It's the exponent of the second term, but that doesn't matter. Eliminate Choice (2).

Choice (4), 4, is the leading coefficient, meaning that it would be the coefficient of the first term if the expression was written in the correct order (standard form). Eliminate Choice (4)

More to come. Comments and questions welcome.

Questions, comments and corrections welcome.

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