Friday, April 20, 2018

Algebra 2 Problems of the Day

Continuing with daily Algebra 2 questions and answers.

More Algebra 2 problems.

January 2018
21. What is the inverse of f(x) = -6(x - 2)?

(1) f-1(x) = -2 - x/6
(2) f-1(x) = 2 - x/6
(3) f-1(x) = 1 / (-6(x - 2))
(4) f-1(x) = 6(x - 2)

Answer: (2) f-1(x) = 2 - x/6
Inverse operations. Divide by negative six, then add two.
x = -6(f-1(x) - 2)
x / (-6) = f-1(x) - 2
2 - x/6 = f-1(x).



22. Brian deposited 1 cent into an empty non-interest bearing bank account on the first day of the month. He then additionally deposited 3 cents on the second day, 9 cents on the third day, and 27 cents on the fourth day. What would be the total amount of money in the account at the end of the 20th day if the pattern continued?
(1) $11,622,614.67
(2) $17,433,922.00
(3) $116,226,146.80
(4) $1,743,392,200.00

Answer: (2) $17,433,922.00
Do not answer the 20th term in the geometric sequence. They are looking for the sum of the first 20 terms.
The formula for finding the sum of the first n terms in a geometric sequence is

Sn = (a1(1 - rn)) / (1 - r),

where n is the number of terms, r is the common ratio, and a1 is the initial term.
In this question, the common ratio is 3, because the sequence goes 1, 3, 9, 27 ...
So the sum is (1 * (1 - 320)/(1 - 3) = 1743392200, which is the number of cents. Divide this by 100 to convert it to dollars, or $17,433,922.00.
Alternatively, you could have used .01 for the initial term in the formula, which would have given you the answer immediately.



Comments and questions welcome.

More Algebra 2 problems.

Thursday, April 19, 2018

Algebra 2 Problems of the Day

Continuing with daily Algebra 2 questions and answers.

More Algebra 2 problems.

January 2018
19. If p(x) = 2x3 - 3x + 5, what is the remainder of p(x) : (x - 5)?

(1) -230
(2) 0
(3) 30
(4) 240

Answer: (4) 240
The Polynomial Remainder Theorem tells us that is p(x) is divided by (x - r), then the remainder, R, can be found by evaluating p(r).
If (x - 5) is a factor of p(x), then when x = 5, p(x) would = 0. If it is not a factor, then the value of p(5) will be the remainder when you divide the polynomials.
If you calculate p(5), you will get 2(5)3 - 3(5) + 5 = 240, which is the remainder.
Alternatively, if you forgot this, you can do the polynomial division. This will give you 240 as a remainder. See the image below:





20. The results of simulating tossing a coin 10 times, recording the number of heads, and repeating this 50 times are shown in the graph below.

Based on the results of the simulation, which statement is false?
(1) Five heads occurred most often, which is consistent with the theoretical probability of obtaining a heads.
(2) Eight heads is unusual, as it falls outside the middle 95% of the data.
(3) Obtaining three heads or fewer occurred 28% of the time.
(4) Seven heads is not unusual, as it falls within the middle 95% of the data.

Answer: (2) Eight heads is unusual, as it falls outside the middle 95% of the data.
Eight does not fall outside the middle 95% of the data. There are 50 data points, so 47.5 pieces of data are in the middle, leaving 2.5 / 2 = 1.25 pieces of data more than two standard deviations above and below the mean. But there are two results greater than 8, so it's not outside of the middle 95%.



Comments and questions welcome.

More Algebra 2 problems.

Wednesday, April 18, 2018

Algebra 2 Problems of the Day

Continuing with daily Algebra 2 questions and answers.

More Algebra 2 problems.

January 2018
17. The function below models the average price of gas in a small town computations. since January 1st.
G(t) = -0.0049t4 + 0.0923t3 - 0.56t2 + 1.166t + 3.23, where 0 ≤ t ≤ 10.

If G(t) is the average price of gas in dollars and t represents the number of months since January 1st, the absolute maximum G(t) reaches over the given domain is about

(1) $1.60
(2) $3.92
(3) $4.01
(4) $7.73

Answer: (3) $4.01
Graph the function and use "maximum" to find the highest value, which you should see is just above $4.00.
See the graph below:

At approximately t = 1.6, G(t) = 4.01, approximately.



18. Written in simplest form, (c2 - d2) / (d2 + cd - 2c2), where c =/= d, is equivalent to
(1) (c + d) / (d + 2c)
(2) (c - d) / (d + 2c)
(3) (-c - d) / (d + 2c)
(4) (-c + d) / (d + 2c)

Answer: (3) (-c - d) / (d + 2c)
The numerator, (c2 - d2), is the difference of two perfect squares, and factors into the conjugates, (c + d)(c - d).
Note that all four choices have (d + 2c) as the denominator, which makes factoring (d2 + cd - 2c2) that much easier into (d + 2c)(d - c).
(c - d) / (d - c) = -1, which reduces the fraction to (-1)(c + d) / (d + 2c).
Distribute the -1, and you get choice (3).



Comments and questions welcome.

More Algebra 2 problems.

Tuesday, April 17, 2018

E-mote

(Click on the comic if you can't see the full image.)

(C)Copyright 2018, C. Burke.

They're irrational, you know.

I remember when they were just ''smileys''. Then ''emoticons'' (emote icons). Finally, ''emoji''. Like ''Gojira'' instead of ''Godzilla''.




Come back often for more funny math and geeky comics.




Algebra 2 Problems of the Day

Algebra 2 is not my usual subject, but I do get asked about the problems occasionally. So I've decided to run a couple of Regents problems daily for a while. If there's a positive reaction (or at least, a lack of negative reaction), I may continue it.

More Algebra 2 problems.

January 2018
15. The terminal side of θ, an angle in standard position, intersects the unit circle at P(-1/3, -sqrt(8)/3). What is the value of sec θ?

(1) -3
(2) -3*sqrt(8)/8
(3) -1/3
(4) -sqrt(8)/3

Answer: (1) -3
The coordinates of P are (cos θ, sin θ)
sec θ = 1 / cos θ
cos θ = -1/3
sec θ = 1 / (-1/3) = -3





16. What is the equation of the directrix for the parabola -8(y - 3) = (x + 4)2?

(1) y = 5
(2) y = 1
(3) y = -2
(4) y = -6

Answer: (1) y = 5
When the parabola is written in this form -- (x − p)2=±4a(y−q) -- then (p,q) will be the vertex and a is the focus length. In other words, the distance in one direction from the vertex will be the focus, and in the other direction will be the directrix.

The vertex is (-4, 3) and the focal length is 2. The negative tells us that the parabola is opening down, so the directrix is 2 units above the vertex, which is y = 5.



Comments and questions welcome.

More Algebra 2 problems.

Monday, April 16, 2018

Algebra 2 Problems of the Day

Algebra 2 is not my usual subject, but I do get asked about the problems occasionally. So I've decided to run a couple of Regents problems daily for a while. If there's a positive reaction (or at least, a lack of negative reaction), I may continue it.

More Algebra 2 problems.

January 2018
13. If aebt = c, where a, b, and c are positive, then t equals

Answer: (3) ln(c/a) / b
You start with: aebt = c
Divide both sides by a: ebt = c/a
Take the natural log: ln(ebt) = ln(c/a)
which gives you: bt = ln(c/a)
Divide by b: t = ln(c/a) / b.



14. For which values of x, rounded to the nearest hundredth, will |x2 - 9| - 3 = log3x?

(1) 2.29 and 3.63
(2) 2.37 and 3.54
(3) 2.84 and 3.17
(4) 2.92 and 3.06

Answer: (1) 2.29 and 3.63
If you graph the system: y = |x2 - 9| - 3 and y = log(x)/log(3), you can use the intersection function the points of intersection (2.29, 0.754) and (3.63, 1.173).
Use (2nd)(CALC), option (5)Intersect and hit ENTER three times.
Or you can graph log(x)/log(3) - |x2 - 9| + 3, and look for the zeroes.
Use (2nd)(CALC), option (2)Zero.

Given that this is a multiple choice question, you could also use a list of information and enter that last equation into the calculator to see which gives you zero -- or very close to zero, because we have approximate answers. Be careful, though, because there's a lot of information to enter and typos happen.



Comments and questions welcome.

Sunday, April 15, 2018

Algebra 2 Problems of the Day

Algebra 2 is not my usual subject, but I do get asked about the problems occasionally. So I've decided to run a couple of Regents problems daily for a while. If there's a positive reaction (or at least, a lack of negative reaction), I may continue it.

More Algebra 2 problems.

January 2018
11. If n = sqrt(a5) and m = a, where a > 0, an expression n/m could be

(1) a5/2
(2) a4
(3) (a2)1/3
(4) (a3)1/2
See image below

Answer: (4) (a3)1/2
The square root of a value is the same as raising it to a power of 1/2, so n can be expressed as a5/2.
Also, m can be expressed as a1.
This means that n/m is the same as (a5/2)/a1.
When dividing, keep the base, subtract the exponents: a(5/2 - 1) = a(3/2)
A fractional exponent of 3/2 would mean take the square root of the third power, which is choice (4).



12. The solutions to x + 3 - (4 / (x - 1) ) = 5 are


Answer: (1) 3/2 + sqrt(17)/2
Follow the logic in the image:


Subtract 5 from each side, then add (4 / (x - 1)) to each side.
This will set up a rational equation. Multiply both sides by (x - 1) -- or "cross-multiply", if you prefer.
Multiply the binomials, then set up the quadratic equation to solve.
Use the Quadratic Formula to find the roots. Note that you have a positive discriminant, so the roots are real, and there is no "i" in the answer.
If you split the file fraction, you get choice (1).



Comments and questions welcome.

Saturday, April 14, 2018

Algebra 2 Problems of the Day

Algebra 2 is not my usual subject, but I do get asked about the problems occasionally. So I've decided to run a couple of Regents problems daily for a while. If there's a positive reaction (or at least, a lack of negative reaction), I may continue it.

More Algebra 2 problems.

January 2018
9. What is the quotient when 10x3 - 3x2 - 7x + 3 is divided by 2x - 1?

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

Answer: (4) 5x2 + x - 3
First of all, if you divide +3 by -1, the result must be -3, so we can eliminate choices (1) and (2).
Since it's multiple choice, it might be easier just to multiply the two remaining choices by 2x - 1 to see which one works. As they only differ by one sign, it should be quick to do, as shown in the image below:


As you can see, +x gives you +2x2, which when added to the -5x2 in the first column, makes a total of -3x2. So the correct answer is choice (4).

If you wanted to divide (after eliminating the two bad choices), 2x - 1 goes into (10x3 - 3x2), 5x2 times.
(10x3 - 3x2) - (10x3 - 5x2) = 2x2
Bring down the next term, -7x. At this point, you will notice that 2x goes into (2x2), +x times, not -x times. You now have enough information to answer the question.



10. Judith puts $5000 into an investment account with interest compounded continuously. Which approximate annual rate is needed for the account to grow to $9110 after 30 years?

(1) 2%
(2) 2.2%
(3) 0.02%
(4) 0.022%

Answer: (1) 2%
You can check each rate to see which gives you $9110 after 30 years, or you can work backward to solve it.
Use the Continuously Compounded Interest formula A = Pert
9100 = 5000e30r
(9100/5000) = e30r
ln(9100/5000) = ln(e30r)
ln(9100/5000) = 30r
ln(9100/5000)/30 = r
r = 0.0199978..., which is approximately 0.02, or 2%.



Comments and questions welcome.

Friday, April 13, 2018

Algebra 2 Problems of the Day

Algebra 2 is not my usual subject, but I do get asked about the problems occasionally. So I've decided to run a couple of Regents problems daily for a while. If there's a positive reaction (or at least, a lack of negative reaction), I may continue it.

January 2018
7. 7 There are 440 students at Thomas Paine High School enrolled in U.S. History. On the April report card, the students’ grades are approximately normally distributed with a mean of 79 and a standard deviation of 7. Students who earn a grade less than or equal to 64.9 must attend summer school. The number of students who must attend summer school for U.S. History is closest to

(1) 3
(2) 5
(3) 10
(4) 22

Answer: (3) 10.
If the standard deviation is 7, then 72 is one standard deviation from the mean, and 65 is two standard deviations from the mean. So anyone who scored 64.9 or lower had a score more than 2 standard deviations away from the mean. That would be approximately 2.3% of the 440 students, which is .023 * 440 = 10.12, or about 10.



8. For a given time, x, in seconds, an electric current, y, can be represented by
y = 2.5(1 - 2.7-10x). Which equation is not equivalent?


Answer: (4)
In each of the four choices, the 25 has been distributed. The Distributive Property gives you choice (1), so eliminate that. In choice (2), you would multiply the exponents (2) and (-.05x) because you have a power of a power, and that gives you -.10x again, so this can be eliminated. Choice (3) removes the negative by inverting the number into its reciprocal, which is what a negative exponent does, so this is eliminated. Choice (4), the only one remaining, shows the product of two expressions with the same base, but the rule would be the add the exponents, not multiply them.

Adding the exponents in choice (4) would yield (-2 + .05x), not (-10x). Choice (4) is not equivalent.



Comments and questions welcome.

Thursday, April 12, 2018

Algebra 2 Problems of the Day

Algebra 2 is not my usual subject, but I do get asked about the problems occasionally. So I've decided to run a couple of Regents problems daily for a while. If there's a positive reaction (or at least, a lack of negative reaction), I may continue it.

More Algebra 2 problems.

January 2018
5. A certain pain reliever is taken in 220 mg dosages and has a half-life of 12 hours. The function A = 220(1/2)t/12 can be used to model this situation, where A is the amount of pain reliever in milligrams remaining in the body after t hours.
According to this function, which statement is true?

(1) Every hour, the amount of pain reliever remaining is cut in half.
(2) In 12 hours, there is no pain reliever remaining in the body.
(3) In 24 hours, there is no pain reliever remaining in the body.
(4) In 12 hours, 110 mg of pain reliever is remaining.

Answer: (4) In 12 hours, 110 mg of pain reliever is remaining.
Substitute 12 for t, and the exponent becomes (12/12), which is 1. 220 times (1/2) is 110 mg. Even without doing any math, you are told in the question that the pain reliever has a "half-life of 12 hours" -- meaning that in 12 hours, there will be half as much, which is 110.
Choice (1) is incorrect because of the fraction in the exponent. Had the exponent been simply t, then Choice (1) would have been correct. Choices (3) and (4) can be eliminated because exponential decay will not go to zero.



6. The expression (x + a)(x + b) can not be written as

(1) a(x + b) + x(x + b)
(2) x2 + abx + ab
(3) x2 + (a + b)x + ab
(4) x(x + a)+ b(x + a)

Answer: (2) x2 + abx + ab
The coefficient for the middle term will be the sum of a and b (as in Choice (3)), not their product.
If you use the Distributive Property on choices (1), (3) and (4), you will get x2 + ax + bx + ab in each case.



Comments and questions welcome.

Twisted System

(Click on the comic if you can't see the full image.)

(C)Copyright 2018, C. Burke.

Each individual line is asymptoted. It's more of a protest song.




Come back often for more funny math and geeky comics.




Wednesday, April 11, 2018

Algebra 2 Problems of the Day

Algebra 2 is not my usual subject, but I do get asked about the problems occasionally. So I've decided to run a couple of Regents problems daily for a while. If there's a positive reaction (or at least, a lack of negative reaction), I may continue it.

More Algebra 2 problems.

January 2018
3. For the system shown below, what is the value of z?


y = -2x + 14
3x - 4z = 2
3x - y = 16

(1) 5
(2) 2
(3) 6
(4) 4

Answer: (4) 4
Substitute -2x + 14 for y in the 3rd equation
3x - (-2x + 14) = 16
3x + 2x - 14 = 16
5x = 30
x = 6
Substitute 6 for x in the 2nd equation
3(6) - 4z = 2
18 - 4z = 2
-4z = -16
z = 4



4. The hours of daylight, y, in Utica in days, x, from January 1, 2013 can be modeled by the equation y = 3.06 sin(0.017x - 1.40) + 12.23. How many hours of daylight, to the nearest tenth, does this model predict for February 14, 2013?
(1) 9.4
(2) 10.4
(3) 12.1
(4) 12.2

Answer: (2) 10.4
February 1 is 31 days after January 1. February 14 is 13 days after February 1. So 31 + 13 = 44.
3.06 sin(0017(44) - 1.40) + 12.23 = 10.3733...
Make sure the calculator is in radians mode, not degree mode (which would give you 12.1952 ... sneaky!)



Comments and questions welcome.