Monday, April 23, 2018

Boyfriend

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(C)Copyright 2018, C. Burke.

Because it is a high school, and these things happen, even when I can't work math into it.




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Algebra 2 Problems of the Day

Continuing with daily Algebra 2 questions and answers.

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January 2018, Part II

Questions in Part II are worth 2 credits. All work must be shown or explained for full credit. A correct numerical answer without work is only worth 1 credit.


27.A formula for work problems involving two people is shown below.

1/t1 + 1/t2 = 1/tb
t1 = the time taken by the first person to complete the job
t2 = the time taken by the second person to complete the job
tb = the time it takes for them working together to complete the job

Fred and Barney are carpenters who build the same model desk. It takes Fred eight hours to build the desk while it only takes Barney six hours. Write an equation that can be used to find the time it would take both carpenters working together to build a desk.
Determine, to the nearest tenth of an hour, how long it would take Fred and Barney working together to build a desk.

Answer:
The equation needed to solve this would be

1/8 + 1/6 = 1/tb or 1/8 + 1/6 = 1/x

Once you have this, you can solve for x (or tb).

3/24 + 4/26 = 1/x
7/24 = 1/x
x = 24/7 = 3.428571... = 3.4 hours

Alternatively, multiply the entire equation by (8)(6)(x) to eliminate the fractions:

(8)(6)(x)(1/8) + (8)(6)(x)(1/6) = (8)(6)(x)(1/x)
6x + 8x = 48
14x = 48
x = 48 / 14 = 3.4 hours.

Also of note, whoever wrote this questions obviously likes The Flintstones.



28.Completely factor the following expression:

x2 + 3xy + 3x3 + y

Answer:
First, rewrite the expression as 3x3 + x2 + 3xy + y
Factor by grouping: x2(3x + 1) + y(3x + 1)
Factor again: (x2 + y)(3x + 1)



Comments and questions welcome.

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Sunday, April 22, 2018

Algebra 2 Problems of the Day (open-ended)

Continuing with daily Algebra 2 questions and answers.

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January 2018, Part II

Questions in Part II are worth 2 credits. All work must be shown or explained for full credit. A correct numerical answer without work is only worth 1 credit.
Elizabeth tried to find the product of (2 + 4i) and (3 - i), and her work is shown below.


(2 + 4i)(3 - i)
= 6 - 2i + 12i - 4i2
= 6 + 10i - 4i2
= 6 + 10i - 4(1)
= 6 + 10i - 4
= 2 + 10i

Identify the error in the process shown and determine the correct product of (2 + 4i) and (3 - i).

Answer:
Elizabeth replaced i2 with 1 instead of -1.
The correct answer is:


= 6 + 10i - 4(-1)
= 6 + 10i + 4
= 10 + 10i




26.A runner is using a nine-week training app to prepare for a “fun run.” The table below represents the amount of the program completed, A, and the distance covered in a session, D, in miles.

Based on these data, write an exponential regression equation, rounded to the nearest thousandth, to model the distance the runner is able to complete in a session as she continues through the nine-week program.

Answer:

Enter the data into two lists (L1 and L2, most likely). Check for errors.
Go to STAT, CALC and select ExpReg.
You should get the following output:
y = a*b^x
a = 1.223034549
b = 2.652024589
Round these numbers to the nearest thousandth. (You will lose a point if you do not round correctly.)
y = 1.223(2.652)x.
You could have used A for x and D for y in your answer.



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Saturday, April 21, 2018

Algebra 2 Problems of the Day

Continuing with daily Algebra 2 questions and answers.

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January 2018
23. If the function g(x) = acx represents exponential growth, which statement about g(x) is false?

(1) a > 0 and b > 1
(2) The y-intercept is (0, a).
(3) The asymptote is y = 0.
(4) The x-intercept is (b, 0).

Answer: (4) The x-intercept is (b, 0).
The function has no x-intercept. And when x = b, then g(x) = abb, not 0.
Note that choice (3) and (4) are mutually exclusive, so one of them has to be false.
Since it is exponential growth, a > 0 and b > 1. And when x = 0, g(x) = ab0 = a.



24.At her job, Pat earns $25,000 the first year and receives a raise of $1000 each year. The explicit formula for the nth term of this sequence is an = 25,000 + (n - 1)1000. Which rule best represents the equivalent recursive formula?
(1) an = 24,000 + 1000n
(2) an = 25,000 + 1000n
(3) a1 = 25,000, an - 1 + 1000
(4) a1 = 25,000, an + 1 + 1000

Answer: (3) a1 = 25,000, an - 1 + 1000
In the recursive formula, each term is the sum of the term before it (an-1) plus 1000, which is choice (3).
Note that choices (1) and (2) are not recursive formulas.



Comments and questions welcome.

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Friday, April 20, 2018

Algebra 2 Problems of the Day

Continuing with daily Algebra 2 questions and answers.

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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.



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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.

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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).



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Tuesday, April 17, 2018

E-mote

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(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.

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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.