## Friday, March 29, 2024

### January 2024 Algebra 2 Regents Part IV

This exam was adminstered in January 2024.

More Regents problems.

### Algebra 2 January 2024

Part IV: A correct answer will receive 6 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.

37.A manufacturer of sweatshirts finds that profits and costs fluctuate depending on the number of products created. Creating more products doesn’t always increase profits because it requires additional costs, such as building a larger facility or hiring more workers. The manufacturer determines the profit, p(x), in thousands of dollars, as a function of the number of sweatshirts sold, x, in thousands. This function, p, is given below.

p(x) = -x3 + 11x2 - 7x - 69

Graph y = p(x), over the interval 0 < x < 9, on the set of axes below

Over the given interval, state the coordinates of the maximum of p and round all values to the nearest integer. Explain what this point represents in terms of the number of sweatshirts sold and profit.

Determine how many sweatshirts, to the nearest whole sweatshirt, the manufacturer would need to produce in order to first make a positive profit. Justify your answer.

Put the function in your graphing calculator and get the Table of Values. Plot the points on the graph from 0 through 9. Connect them but DO NOT add any arrows that go beyond the specified range. Notice that each box on the y-axis is 15 units, while each box on the x-axis is one.

The points you need to plot are (0,-69), (1,-66), (2,-47), (3,-18), (4,15), (5,46), (6,69), (7,78), (8,67), (9,30).

Your graph will look like this:

The coordinates maximum value, to the nearest interger, are (7,78). This means that for 7,000 sweatshirts the profit will be \$78,000.

Using the calculator, you can find the zero of the function to be (3.5488,0). Since the function represents thousands of sweatshirts, then this is 3548.8 sweatshirts. That means that the first positive profit will be at 3549 sweatshirts.

End of Exam

How did you do?

More to come. Comments and questions welcome.

More Regents problems.

### I also write Fiction!

You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

## Monday, March 25, 2024

### Positive One

(Click on the comic if you can't see the full image.)
(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Like your imaginary friends, you can be the positive one.

Now, I know the value of i, but what do we know about all those b g's?

If you're wondering why I didn't do a "four i" joke, well, because I did one a looooong time ago, in January 2011.

This wasn't a good "Mike" comic and definitely not a student comic, so I fell back on the tried and true stick figures.

A little trivia: I've lived a good part of the past three decades only a few blocks from that famous street walk from Saturday Night Fever. The famed Lenny's Pizza just closed its doors recently after all those years.

### I also write Fiction!

You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Come back often for more funny math and geeky comics.

## Thursday, March 14, 2024

### Pi Day 2024

(Click on the comic if you can't see the full image.)
(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

The co-sin as well!

I know I usually save these guys for Talk Like A Priate Day, and use Sherlock Pi on Pi Day (3/14) but I decided to shake things up a little.

I've been getting a lot of suggestions for using the Sin of Pie is 0, which is amusing, but it's not my joke, so unless I can work it into my characters, I can't use it. (I could, but I choose not to.)

On the other hand, if you quarter pi, then the sin is radical 2 over 2. And so is the cos.

And your tan will be the one.

### I also write Fiction!

You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Come back often for more funny math and geeky comics.

## Tuesday, March 12, 2024

### January 2024 Algebra 2 Regents, Part III

This exam was adminstered in January 2024.

More Regents problems.

### Algebra 2 January 2024

Part III: Each correct answer will receive 4 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.

33. A researcher wants to determine if nut allergies and milk allergies are related to each other. The researcher surveyed 1500 people and asked them if they are allergic to nuts or milk. The survey results are summarized in the table below.

Determine the probability that a randomly selected survey respondent is allergic to milk
Determine the probability that a randomly selected survey respondent is allergic to milk, given that the person is allergic to nuts.
Based on the survey data, determine whether nut allergies and milk allergies are independent events. Justify your answer.

Add up the rows and columns. You will see that that it adds to 1500, as stated in the problem.

There are 45 respondents out of 1500 who are allergic to milk, so the probability is 45/1500. (You don't need to simplify the fraction.)

There are 15 people who are allergic to nuts, and of those, 3 are also allergic to milk, so the probability is 3/15.

If the nut allergies and milk allergies are independent, then the previous two answers would be the same because P(A) would have to be equal to P(A|B). However, 45/1500 = 0.03 and 3/15 = 0.2. So the events are not independent.

34. Algebraically solve for x: 2x = 6 + 2√(x - 1)

Isolate the radical. Then square both sides. Finally, solve the quadratic equation.

2x = 6 + 2√(x - 1)
2x - 6 = 2√(x - 1)
x - 3 = √(x - 1)
(x - 3)2 = (√(x - 1))2
x2 - 6x + 9 = x - 1
x2 - 7x + 10 = 0
(x - 5)(x - 2) = 0
x - 5 = 0 or x - 2 = 0
x = 5 or x = 2

Throw out x = 2 as extraneous because 2(2) =/= 6 + 2√(2-1).

The only solution is x = 5.

35. During the summer, Adam saved \$4000 and Betty saved \$3500. Adam deposited his money in Bank A at an annual rate of 2.4% compounded monthly. Betty deposited her money in Bank B at an annual rate of 4% compounded quarterly. Write two functions that represent the value of each account after t years if no other deposits or withdrawals are made, where Adam’s account value is represented by A(t), and Betty’s by B(t).

Using technology, determine, to the nearest tenth of a year, how long it will take for the two accounts to have the same amount of money in them. Justify your answer.

Write the functions A(t) and B(t) using the given initial amounts and rates. Note that that A is compounded monthly, so 0.024 will be divided by 12 and the exponent will be multiplied by 12. Likewise, B is compounded quarterly, so 0.04 will be divided by 4 and the exponent will be multiplied by 4.

A(t) = 4000(1 + 0.024/12)12t
or A(t) = 4000(1.002)12t

B(t) = 3500(1 + 0.04/4)12t
or B(t) = 3500(1.01)4t

"Using technology" means that you can graph the two functions to see when they intersect rather than writing an equation and solving. Put the two equations into your graphing calculator. Graph and trace the functions, or look at the table of values, setting the calculator to show every 0.1 value of x.

At t=8.4, A(8.4) = 4892.40 and B(8.4) = 4889.50, a difference of \$3.10, which is the smallest difference to the nearest tenth of year.

36.
On the graph below, draw at least one complete cycle of a sine graph passing through point (0,2) that has an amplitude of 3, a period of p, and a midline at y = 2.
Based on your graph, state an interval in which the graph is increasing.

The period of π instead of 2π means that means we need sin(2x) instead of sin(x). The amplitude of 3 is a mulitplier in front of the function and the midline of 2 is an addition after the function.

y = 3 sin(2x) + 2

The graph would look like the one below. The y-intercept is at (0,2). The maximum value is 5 and the minimum is -1.

One interval where it is increasing would be from 3π/4 to 5π/4.

End of Part III

How did you do?

More to come. Comments and questions welcome.

More Regents problems.

### I also write Fiction!

You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

## Friday, March 08, 2024

### Arctan

(Click on the comic if you can't see the full image.)
(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

I know, the Ham joke was a little cheesy.

I haven't used Noah and his sons in quite a while. They were one of the early themes, and I thought I'd revisit them as we closed in on comic #2000.

Also, as anyone with fair skin can tell you, you can get burned on cloudy days. UV rays don't care about clouds.

Since I have such fair skin, I tend to stay covered up even in the summer, so I have more of an inverse tan.

### I also write Fiction!

You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Come back often for more funny math and geeky comics.