Social Media Logic Leap

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(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

"And you know what they're really saying..."

It's not just social media, but it's bad there. You take one sentence or a two-second clip, and then you stretch it, distort it and refute the distortions.

I've seen a two-second clip with commentary like, "Really? Does that mean that X? Won't that lead to Y and Z?" And they're on a roll. All this despite the fact that if you look at a seven-second clip or - Heaven Forbid! - 30 seconds or the complete response, it's quite obvious that, no, it doesn't mean X, Y, or even Z, whatever those stand for.

And even if those points aren't addressed, no one will ever ask. They'll just accuse because that's easier.

That's why I do math and apply actual logic. I prefer the challenge.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My older books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Come back often for more funny math and geeky comics.

You're Back!

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(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

"For how long, Mr. Spock? ... For how long?"

I've made promises, and I've broken them. But since I don't know how many people are still reading blogs these days, I can't tell how many people, if any, that I've disappointed. Few have inquired about my extended absences, so it leaves me wondering.

Two things spurred me on: I really want to make it to 2,000 comics, which I should have surpassed a long time ago, and I got an email from Fables dot pro about hosting my comic. Fables got my name from the old TopWebComics site, which still exists, actually, even if I haven't visited it in over a decade. (By contrast, I still check out the Comic Genesis site a few times per year, even if I haven't updated it. And I'm not even sure I remember how to update it. I might need my old PC where there's a saved password.)

More to come. For real this time.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My older books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Come back often for more funny math and geeky comics.

Algebra 2 Problems of the Day (Algebra 2 Regents, August 2024 Part IV)

This exam was adminstered in August 2024 .

August 2024 Algebra 2, Part IV

A correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.

37. Taylor wants to open an investment account with the $1200 she received for her birthday. She has narrowed her choices down to two banks. America’s Bank offers 6.4% annual interest compounded quarterly. Barnyard Bank offers 6.35% annual interest compounded continuously.

Write functions for A(t) and B(t) to represent the value of her investment with America’s Bank and Barnyard Bank as a function of time, t, in years.

Taylor would like to invest the $1200 into one bank for ten years making no additional deposits and no withdrawals. With which bank will Taylor earn the most money? Justify your answer.

Taylor chooses to invest her money in Barnyard Bank. Algebraically determine how long, to the nearest tenth of a year, it will take her initial investment to triple assuming she makes no deposits or withdrawals.
Justify your answer.

Answer:

Compounded quarterly means four times a year. Compounded continuously means you have to use e.

A(t) = 1200(1 + 0.064/4)^4t
B(t) = 1200e^0.0635t

A(10) = 1200(1 + 0.064/4)⁴⁽¹⁰⁾ = 2264.277...
B(t) = 1200e^0.0635(10) = 2264.426...

After 10 years, Barnyard Bank will earn more money.

Triple $1200 is $3600. Some for B(t) = 3600.

3600 = 1200e^0.0635t
3 = e^0.0635t
ln 3 = 0.0635t
ln 3 / 0.0635 = t

t = 17.301...

It will take 17.3 years for the investment to triple.

End of Exam

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My older books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra 2 Problems of the Day (Algebra 2 Regents, August 2024 Part III)

This exam was adminstered in August 2024 .

August 2024 Algebra 2, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.

33. The population of China, in millions, can be modeled by the function P(x) = 316.93e^0.0133x , where x is the number of years since 1900. The population of India since 1900 is summarized in the table below:

Which country’s population had a greater average rate of change between 1950 and 2020? Justify your answer.

Answer:

Find the rate of change for India by subtracting the population in 1950 from the population in 2020, and divide it by 70 (years). Find the change for China by evaluating the function for x = 50 and x = 120 (years since 1900) and divide that by 70. Which is bigger?

Note that you can solve the problem without dividing each number by 70, but they specifically asked about the rates, so divide by 70.

India: (1380 - 376.3) / 70 = 1003.7 / 70

China: P(50) = 316.93e^0.0133(50) = 616.267
P(120) = 316.93e^0.0133(120) = 1563.498
(P(120) - P(50)) / 70 = (1563.498 - 616.267) / 70 = 947.231 / 7

India had a greater rate of change because 1003.7 / 70 > 947.231 / 70.

34. In a packaging plant, a machine packs boxes with jars. The machine’s manufacturer states that a box is packed, on average, every 42 seconds. To test that claim, the packaging plant randomly selects a sample of 10 boxes and finds the sample mean to be 49.8 seconds.

The company ran a simulation of 1000 trials based on the manufacturer's claim. The approximately normal results are shown below.

Based on the simulation, determine an interval containing the middle 95% of plausible mean times. Round your answer to the nearest hundredth.

Is the time 49.8 seconds unusual? Use statistical evidence to justify your answer.

Answer:

The middle 95% is the mean minus 2 times the standard deviation to the mean plus 2 times the standard deviation.

42.029 - 6.21 = 35.819, 42.029 + 6.21 = 48.239. (35.82, 48.24) to the nearest hundredth.

The value 49.8 is unusual because it falls outside the middle 95%. (It is greater than 48.24.)

35. Consider the function f(x) = 2^x.
Is f(x) an even function? Justify your answer.

Write an equation for g(x), the function that results after f(x) is shifted up 5 units.

Write an equation for h(x), the inverse of g(x).

Answer:

The function f(x) is not an even function because f(x) =/= f(-x). For example f(1) = 2, f(-1) = 1/2.

g(x) = f(x) + 5 = 2^x + 5

Note: I checked the official response guide and "f(x) + 5" was acceptable as an answer.

To find the inverse, replace x with y, and set the function equal to y. Then solve for x.

x = 2^y + 5
x - 5 = 2^y
log₂ (x - 5) = y

h(x) = log₂ (x - 5)

36. Solve the system of equations shown below algebraically:

(x - 4)² + (y - 1)² = 9
x - y = 6

Answer:

Find the two points on the line that intersect the circle.

Solve for x and substitute in the circle equation.

x - y = 6
x = y + 6

(x - 4)² + (y - 1)² = 9
(y + 6 - 4)² + (y - 1)² = 9
(y + 2)² + (y - 1)² = 9
y² + 4y + 4 + y² - 2y + 1 = 9
2y² + 2y + 5 = 9
2y² + 2y - 4 = 0
y² + y - 2 = 0
(y + 2)(y - 1) = 0
y + 2 = 0 or y - 1 = 0
y = -2 or y = 1

If y = -2, x - (-2) = 6, then x = 4. (4, -2)

If y = 1, x - 1 = 6, then x = 7. (7, 1)

End of Part III

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My older books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra 2 Problems of the Day (Algebra 2 Regents, August 2024 Part II)

This exam was adminstered in August 2024 .

August 2024 Algebra 2, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.

25. On the axes below, graph y = 3.2(1.8)² .

Answer:

Put the equation in your graphing calculator and check the Table of Values.

Points on the graph include: (0,3.2), (1,5.76), (2,10.368), (3,18.6624), (-1, 1.78), (-2,.99), (-3,.55), (-4,.3), (-5,.16). Approxminate these points.

26. Is x + 3 a factor of 7x³ + 27x² + 9x - 27?
Justify your answer.

Answer:

You can show if it is a factor or not by doing doing polynomial division. However, an easier method is this: if (x + 3) is a factor, then the polynomial will equal zero when x = -3.

7(-3)³ + 27(-3)² + 9(-3) - 27 = 0, so (x + 3) is a factor of the polynomial.

27. Over the set of integers, factor the expression 2x⁴ - 10x³ + 3x² - 15x completely.

Answer:

Factor completely means that there will be multiple steps. Over the set of integers means that you don't have to worry about imaginary numbers.

2x⁴ - 10x³ + 3x² - 15x
x(2x³ - 10x² + 3x - 15)
x(2x²(x - 5) + 3(x - 5))
x(2x² + 3)(x - 5)

28. The monthly unemployment rate of towns in the United States is approximately normally distributed with a mean rate of 5.2% and a standard deviation of 1.6%. Determine the percentage of towns, to the nearest integer, that have a monthly unemployment rate greater than 6%.

Answer:

Use the "normalcdf" function on your graphing calculator.

Use the command normalcdf(6,100,5.2,1.6) to find the percentage between 6% and 100% ("greater than 6%), mean 5.2%, standard deviation 1.6%.

The result should be .3085, which is .31, or 31%.

29. The function d(t) = 2cos(π/6 t) + 5 models the water depth, in feet, at a location in a bay, t hours since the last high tide. Determine the minimum water depth of the location, in feet, and justify your answer.

Answer:
The midline is 5 feet (from the +5). The amplitude is 2 feet (from 2cos). This means that the water depth varies between 3 feet and 7 feet.

Therefore, the minimum water depth is 3 feet.

Remember to justify it. This is easy enough to do in your head. Do NOT just write 3 feet, or 5 - 2 = 3. Indicate where the 5 and 2 came from and what they stand for.

30. A brewed cup of coffee contains 130 mg of caffeine. The half-life of caffeine in the bloodstream is 5.5 hours. Write a function, C(t) to represent the amount of caffeine in the bloodstream t hours after drinking one cup of coffee

Answer:
The function is expoonential, with 130 as the initial amount and 1/2 as the rate (half-life). It takes 5.5 hours for 1/2 to leave the bloodstream, but t is measured in hours, so we need to divide t by 5.5.

C(t) = 130(1/2)^t/5.5

31. Markus is a long-distance walker. In one race, he walked 55 miles in t hours and in another race walked 65 miles in t + 3 hours. His rates are shown in the equations below.

r = 55/t ; r = 65 / (t + 3)

Markus walked at an equivalent rate, r, for each race. Determine the number of hours that each of the two races took.

Answer:
Write a proportion and solve it.

55/t = 65/(t + 3)
65t = 55(t + 3)
65t = 55t + 165
10t = 165
t = 16.5

The first race took 16.5 hours, and the second race took 19.5 hours.

32. Solve the equation x² + 3x + 11 = 0 algebraically. Express the answer in a + bi form.

Answer:
Use the Quadratic Formula to solve this.

x = (-b + √(b² - 4ac) / (2a)
x = (-3 + √((3)² - 4(1)(11)) / (2(1))
x = (-3 + √(9 - 44)) / (2)
x = (-3 + √(-35)) / (2)
x = -3/2 + √(35)/2 i

Note that you MUST separate the expression into two fraction because the question states a + bi form.

End of Part II

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My older books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Geometry Problems of the Day (Geometry Regents, August 2024 Part IV)

This exam was adminstered in August 2024 .

August 2024 Geometry, Part IV

A correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.

35. In quadrilateral HOPE below, EH ≅ OP, EP ≅ OH, EJ ≅ OG, and TG and YJ are perpendicular to diagonal EO at points G and J, respectively.

Prove that TG ≅ YJ.

Answer:

This is a long proof. Look at what you are given and what you need to find.

To show that TG = YJ, you need to show that triangle GET ≅ triangle JOY. To show that, you need to use one of the triangle postulates or theorems.

You are given that there are perpendicular lines, which create right angles, and we can show that there's a parallelogram so there's another angle. That means we could do with with either ASA or AAS.

Statement	Reason
1. EH ≅ OP, EP ≅ OH, EJ ≅ OG, and TG and YJ are perpendicular to diagonal EO at points G and J, respectively.	1. Given
2. HOPE is a parallelogram	2. A quadrilateral with two pairs of congruent sides is a parallelogram.
3. Angle JOY ≅ angle GOT	3. Alternate interior angles
4. Angle EGT and angle OJY are right angles.	4. Definition of perpendicular lines
5. GJ ≅ GJ	5. Reflexive Property
6. GE ≅ JO	6. Subtration Postulate
7. Triangle GET ≅ Triangle JOY	7. ASA Postulate
8. TG ≅ YJ.	8. CPCTC

End of Part Exam

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My older books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Geometry Problems of the Day (Geometry Regents, August 2024 Part III)

This exam was adminstered in August 2024 .

August 2024 Geometry, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.

32. A drone is used to measure the size of a brush fire on the ground. Segment AB represents the width of the fire, as shown below. The drone calculates the distance to point B to be 1076 feet at an angle of depression of 25°. At the same point, the drone calculates the distance to point A to be 774 feet at an angle of depression of 36°.

Determine and state the width of the fire, AB, to the nearest foot.

Answer:

Call the right angle C. To find the length of AB, you need to find the lengths of AC and BC and then subtract.

Because the two horizontal lines shown in the picture are parallel (all horizontal lines have slope = 0 and are therefore parallel), then we know that Angle A has a measure of 36° and Angle B has a measure of 25°.

In both cases, we have the hypotenuse and we are looking for the adjacent side of the right triangle. (The opposite side is the height. We don't know it, and we are not looking for it.) This means we must use cosine. Twice.

Cos 25 = x / 1076
x = 1076 cos 25 = 975.1871...

Cos 36 = x / 774
x = 774 cos 36 = 626.1791...

975.1871 - 626.1791 = 349.008

The length of AB is 349 feet.

33. 3 Quadrilateral ABCD has vertices with coordinates A(-3,6), B(6,3), C(6,-2), and D(-6,2). Joe defines an isosceles trapezoid as a trapezoid with congruent diagonals. Use Joe’s definition to prove ABCD is an isosceles trapezoid. [The use of the set of axes below is optional.]

Answer:

You don't have to graph the figure but it will make the problem easier to visualize. Also, you must show that this is an isosceles trapezoid using Joe's method or you will not receive full credit.

First, show that the figure is a trapezoid by showing that there is only one pair of parallel sides with the same slope. The slope of AB is -1, and the slope of CD is -1, so AB || CD. The slope of BC is undefined (a vertical line), and the slope of DA is 4/3, so BC is NOT parallel to DA. Therefore, ABCD is a trapezoid.

Second, in a trapezoid, the diagonals are congruent. so use the Distance Formula (or Pythagorean Theorem) to find the lengths of diagonals AC and BD.

AC = &sqrt;( (-3 - 6)² + (6 - -2)²) = &sqrt;(81 + 64) = &sqrt;(145)

BD = &sqrt;( (-6 - 6)² + (3 - 2)²) = &sqrt;(144 + 1) = &sqrt;(145)

Since AC ≅ BD the trapezoid is isosceles.

If you found that AD is congruent to BC, then you would not get full credit for this problem.

34. Ali made six solid spherical decorations out of modeling clay. Each decoration has a radius of 2.5 inches. The weight of clay is 68 pounds per cubic foot.

Determine and state, to the nearest pound, the total weight of the six decorations.

Answer:

You are given the weight per cubic foot. You need to find the number of cubic feet total in all six decorations. However, you are given measurements in inches, so you'll need to divide by 12³.

V = 4/3 π r³ = 4/3 π (2.5)³ = 65.4498 ...

Six decorations have a volume of 6 * 65.4498 = 392.6990 cubic inches or 392.6990/12³ = 0.227 cubic feet

The weight is 0.227 * 68 = 15.436 or 15 pounds.

End of Part III

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My older books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!