Tuesday, November 13, 2018

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

June 2017, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


19. To solve,

Ren multiplied both sides by the least common denominator. Which statement is true?
(1) 2 is an extraneous solution
(2) 7/2 is an extraneous solution
(3) 0 and 2 are extraneous solutions
(4) This equation does not contain any extraneous solutions.

Answer: (1) 2 is an extraneous solution.
The equation is undefined for 0 and 2 because of the terms in the denominator. When both sides are multiplied by x2 - 2x, that restriction is lifted. If either 0 or 2 are solutions to the equation, then they are extraneous.
Look at the steps below:

At this point, you can check both 0 and 2 to see if they are solutions to the equation.
2(0)2 - 11(0 - 2) = 0 - 11(-2) = 22, which is not equal to 8. Zero is not an extraneous solution.
2(2)2 - 11(2 - 2) = 2(4) - 11(0) = 8 - 0 = 8. Two is an extraneous solution.
As for 7/2, if you finish working out the problem, you will see that it is a solution to the equation, but it is not an extraneous solution.





20. Given f(9) = -2, which function can be used to generate the sequence -8, -7.25, -6.5, -5.75,...?
(1) f(n) = -8 + 0.75n
(2) f(n) = -8 - 0.75(n - 1)
(3) f(n) = -8.75 + 0.75n
(4) f(n) = -0.75 = 8(n - 1)

Answer: (3) f(n) = -8.75 + 0.75n
The rate of change is +0.75, which eliminates (2) and (4).
Choice (1) is incorrect because it has an incorrect starting value. If you substitute n = 1, you get -7.25 instead of 8.
Choice (2) would have been correct if there had been addition instead of subtraction.
Choice (3) is the correct answer because adding 0.75 to -8.75 gives you the initial term of -8.





21. The function

represents a damped sound wave function. What is the average rate of change for this function on the interval [-7,7], to the nearest hundredth?
(1) -3.66
(2) -0.30
(3) -0.26
(4) 3.36

Answer: (3) -0.26
To find the average rate of change, calculate f(7) - f(-7) and divide it by (7 - -7). (Like the slope formula.)
This gives you the following equation:


When you plug this into a calculator -- use a lot of parentheses if you have an older operating system! -- you will get -0.261492..., which is -0.26 to the nearest hundredth.



Comments and questions welcome.

More Algebra 2 problems.

Monday, November 12, 2018

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

June 2017, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


16. For x ≠ 0, which expressions are equivalent to one divided by the sixth root of x?


Which explanation is appropriate for Miles and his dad to make?
1) I and II, only
2) I and III, only>
3) II and III, only
4) I, II, and III

Answer: 4) I, II, and III
"One divided by the sixth root of x" would be the same as "the negative sixth root of x".
The sixth root can be expressed with the fractional exponent 1/6.
Therefore, III is correct.
Again, keeping in mind that the sixth root is exponent 1/6, and third root is exponent 1/3, you can see that choices I and II are both the same. So either they are both correct, or they are both incorrect. This means that choice 4 is the answer.
However, let's show our work:
Because we are dividing the same base, we can subtract the exponents (1/6) - (1/3) = (-1/6). This means that choices I and II are equivalent to III, which is correct, so all three are correct.





17. A parabola has its focus at (1,2) and its directrix is y = -2. The equation of this parabola could be
1) y = 8(x + 1)2
2) y = 1/8(x + 1)2
3) y = 8(x - 1)2
4) y = 1/8(x - 1)2

Answer: 4) y = 1/8(x - 1)2
The standard form of a parabola is y = (1/(4p))(x - h)2 + k, where the focus is (h, k + p) and the directrix is y = k - p. The vertex is (h, k), directly in the middle, and that must be (1, 0), which makes h = 1, k = 0 and p = 2. That makes 1/(4p) = 1/(4*2) = 1/8. Choice 4.





18. The function p(t) = 110e0.03922t models the population of a city, in millions, t years after 2010. As of today, consider the following two statements: I. The current population is 110 million. II. The population increases continuously by approximately 3.9% per year. This model supports
1) I, only
2) II only
3) both I and II
4) neither I nor II

Answer: 2) II only
The rate is given as 0.03922, which is approximately 3.9%.
However, 110 million was the population in 2010, not the "current" population -- which is odd phrasing since no specific time was given. Were students to assume 2017, and seven years after 2010? Thankfully, the Regents do not (usually?) use imprecise questions like this when the information matters!



Comments and questions welcome.

More Algebra 2 problems.

Sunday, November 11, 2018

Happy Veterans Day -- Armistice + 100

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(C)Copyright 2018, C. Burke.

Ten years ago, I wondered ten years ago on 11-11-08, just how big a remembrance there would be this year, on the 100th anniversary.

For anyone counting, there are 100 poppies on that field. I originally started placing them in a Fibonacci spiral, but then abandoned that idea.

To all the veterans out there, Thank You.




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Friday, November 09, 2018

Plan A for Anniversary

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(C)Copyright 2018, C. Burke.

It works out that by the time the house is empty on its own, you don't feel the need to leave it to have a good time.




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Wednesday, November 07, 2018

Another Hour

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(C)Copyright 2018, C. Burke.

Odds are no actual sleep took place within that final hour. Or the snooze afterward.




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Monday, November 05, 2018

How Much Candy?

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(C)Copyright 2018, C. Burke.

N = 0 is also a possibility ... but not very likely.




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