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Week 13 in Pre-Calc 11

This week, I’ve learned about absolute values function on graphing.

And in this blog, I’m going to talk about how to graph an absolute value function of linear equations.

 

Just a recap, but we do know that for absolute values, we must remember that:

  • the sum inside the absolute value symbols will always be positive !

 

Taking the main idea that we can never get any negative answer in absolute values, since we have an equation of y=|mx+b| then if we graph our linear equation, we must never have any lines within the negative y side of the graph.

So how exactly do we graph it?

 

First of all, we must know what the parent function is of the graph equation we are given.

For example, we have y = |3x + 6|

If we take the absolute value out, we get y = 3x + 6

 

Secondly, we must figure out what the x-intercept is, doing it by graphing or algebraically doesn’t matter as long as we figure out what the x-intercept is accurately.

And in our equation y = 3x + 6, we want the y to equal zero so we could figure out our x-intercept.

In this case, it’s

0 = 3x + 6

-6 = 3x

-2 = x

 

Now that we’ve figure out what the x-intercept is, we can start graphing by taking the y-intercept, which is (0, 6) in our equation.

After that, make a vertical dashed line to where the x-intercept is.

Then, graph the equation without the absolute value.

Then, what you’re going to do is to make the graph “reflect” from the x-axis from where the x-intercept is. Remember that the reflected graph HAS THE NEGATIVE SLOPE OF YOUR ORIGINAL LINE!!!

 

If you’re stuck, just remember the idea of absolute value not having a negative as an answer!!

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Wave Interference Activity

Constructive Interference – when two crests/troughs from two sources meet, the energies combine to form a larger wave.

 

Destructive Interference – when a crest and a trough from two sources meet, the energies are against each other, leading to the waves cancel out each other.

 

Standing Wave – when two interfering waves have the same wavelength and amplitude, the result is that the interference wave pattern remains the same or stationary. The point where the wave is at rest is called a node.

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Exploring Waves

Pulse Wave – A single disturbance that moves from one point to another.

 

Periodic Wave – series or repeating of many disturbance at regular, even intervals.

 

Transverse Waves – the displacement of the highest point (crest), or the lowest point (trough) from the mid point of the wave is perpendicular (90° angle) to the direction the wave is travelling to.

 

Longitudinal Wave – The disturbance is in the same direction of the wave it will travel to.

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Archimedes Challenge

History of Compound Pulley System

The compound pulley system is adapted from the original pulley system – one fixed and one moving pulleys.

 

The pulley system has an unknown origin, but in 1500 BC, Mesopotamians used rope pulleys for hoisting water. Think of getting water from wells…

 

The first usage of the compound pulley system can be traced back to Archimedes – recorded by Plutarch, a Greek biographer. Some are almost certain that they used it to build the Stonehenge in the UK.

What happened is that Archimedes used many compound pulleys to move an entire warship using his own strength, which is now known as the claw of Archimedes.

 

So, how exactly did he do that?

It’s because the reason why the pulley systems are built – they are designed to lift great weights using lesser force needed. They are built mainly to get water from wells and lifting heavy objects.

Now, compound pulley systems are used to lift elevators, they are also used in boats and cranes, and there are even gym equipments that use compound pulley systems!

 


Physics Involved in Compound Pulley System

The physics behind the pulley is that, it reduces the amount of force needed to lift the object depending on how many pulleys that you have, and the amount of weight the object you’re lifting has.

 

Now, as stated earlier, the compound pulley system has two parts:

  • a fixed pulley.
  • a moving pulley.

In our project, we’ve decided to have one fixed pulley and one moving pulley.

So, let’s divide this into two parts.

 

Fixed Pulley

There’s nothing much going on in here except for the advantage that you have – you can use your body’s weight to add to the force using to lift the weight,

 

Movable Pulley

This is where most of the physics is going on. The pulley is being supported by two ropes. Since that’s the case, the amount of force that we need to apply to lift the weight is half the object’s weight for each pulley.

 

Compound Pulley

Since we only have one of each pulley types, we can say that

WEIGHT = GRAVITATIONAL FORCE; and GRAVITATIONAL FORCE = mg

 

F = \frac{mg}{P}

F = force applied to pull the rope.

m = object’s mass

g = acceleration due to gravity. (9.81 m/s2 on Earth)

P = number/amount of pulleys

In our project’s situation,

what we can say is:

F = \frac{mg}{2}

with mass being 0.150 kg, and since we have 2 pulleys:

 

F = \frac{(0.150)(9.81)}{2}

 

F = 0.736 N

This might be a minuscule example, but it’s certainly useful, especially the ones that are really stable and with many pulleys, because you’d be able to lift great weights without exhausting a lot of force.

 


Design and Building Process

To not confuse people, we wanted to build an ancient flamethrower initially. However, we met a mishap in the middle of building it due to lack of information of materials needed. I have to admit that it was a fault on our part. So, we had to build a compound pulley system instead.

 

Day 1 + Day 2

We were deciding which machine we would build, and we had agreed on the ancient flamethrower as it piqued our interest.

Then, we started researching on how to build it, the materials needed, and which flamethrower we should base our project into.

 

Day 3

We tried to build it but as we still didn’t have enough information we couldn’t figure out how it would work, sadly. So what we did is to research more about it, and as well as the physics behind it. Nothing too time-efficient, unfortunately.

 

Day 4

Seeing that we were anxious as how we were not much efficient in building the flamethrower, we decided to gamble and researched about the compound pulley system instead. We were able to plan it out smoothly this time, but our only challenge is time. So, we decided to search everything that we would include in our blogs and our design for our machine.

 

Day 5

On day 5, we had our materials and built our machine, but we messed up at some point and had to restart building it.

 

This is the step-by-step of how we built it:

  • made a glider for our moving pulley just at the right height for the frame of our machine.
  • drilled into the center of the movable pulley so we could attach the lock.
  • put on the string to our pulleys (which are yo-yo’s).
  • attach the second rope for our moving pulley at the roof of our frame.
  • glued the fixed pulley on the top of the frame.

References

Ronan Industrial

Groovy Lab in a Box

New World Encyclopedia

Pulley India

http://www.engineeringexpert.net/Engineering-Expert-Witness-Blog/tag/gravity

https://en.wikipedia.org/wiki/Mechanical_advantage_device

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Week 11 in PreCalc 11

Last week in Pre-calculus was the 11th week. Sorry for the one-week delay.

 

One thing that I’ve learned is solving linear inequalities in one variable.

 

Overall, this topic should be quite straightforward. You just need to be careful of your signs (+/-) and the inequality that you need to use.

 

First of all, since we’re only solving in one variable, we don’t need to use the x-y graph, just a number line would be fine.

 

These are all the inequality signs that we’re going to use:

Keep in mind that if you graph them on the number line, there’s a slight difference between greater/less than and greater/less than or equal

 

If you graph an inequality with greater than or less than signs, the number that it’s based on is NOT shaded or part of the solutions.

On the other hand, if you graph it with greater than or equal or less than or equal signs, the number it’s based on IS shaded or part of the solutions…

(greater/less than – left) (greater/less than or equal – right)

 

NOTE: keep in mind that if you DIVIDE or MULTIPLY by a NEGATIVE, the inequality sign must be SWITCHEDcheck example 2!!

Examples:

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Week 12 in PreCalc 11

We dealt with a lot of things this week, especially about systems. And for this blog, I’m going to talk about linear-quadratic system.

 

Let’s have a quick recap:

  • Linear equation deals with a straight line on the graph. Basically, an equation of a line.
    • The equation being known as y = mx + b, where is the slope, and is the y-intercept.
  • Quadratic equation deals with a parabola on a graph, with the equation having at least one squared variable.
    • The standard equation being known as y = a (x – p)2 + qFind out more on my blog post about quadratic equations!
  • Together, they form a relation called System of Linear and Quadratic Equation. 

 

Systems (where they intersect) of these two equations can be find out or solved:

  • graphically.
  • algebraically.
    • substitution.
    • elimination.

However, in this blog I’m only going to talk about solving it graphically.

(At this point, you should be able to graph both of them!)

 

NOTE: There are three possible cases that can happen if you finished graphing them!

(graphs used are in courtesy of desmos!)

  • Two solutions – if the line passes on two points of the parabola
  • One solution – if the line passes only on one point of the parabola.
  • No solution – if the line and the parabola doesn’t cross at all.
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Week 10 in PreCalc 11

As our math midterm is getting near, this week has been a review week which has been a great refresher for my memory. It was good and bad to know that I’ve forgotten a lot of things. But it’s a good thing to reflect to when studying!

Not much was learned, but rather a lot of refreshers. Here are some of them:

I wouldn’t go into much details about them all as I’ve posted them already in my edublog, but I would do some examples. Here are the links to the blog posts:

 

Absolute Values:

  • how far away a number from zero is in a number line.
  • \sqrt{x}^2
  • The answer will always be positive inside the absolute value sign ” | | “
  • Can act like a parenthesis.

Examples:

| 5 – 7 |

= | -2 |

= 2

 

-|-4|

= – | 4

= -4

 

Arithmetic Sequence:

  • A sequence of numbers that are added or subtracted by the same value.
  • Addition of a number sequence. E.g. 1 + 2 + 3 + 4 + 5 + 6….
  • S_n = \frac{n}{2}(a + t_n)
  • S_n = \frac{n}{2}(2a + (n-1)d)
  • n = the number/amount of terms you’re calculating
  • a = the first term
  • tn = last term
  • d = common difference

Example:

2 + 5 + 8 + 11…

n = 20

a = 2

 

Since we don’t have tn, we’ll be using the second formula.

S_n = \frac{n}{2}(2a + (n-1)d) S_{20} = \frac{20}{2}(2(2) + (20-1)3) S_{20} = 10 (4 + 57) S_{20} = 10 (61) S_{20} =610

 

Adding and Subtracting Radicals

  • add/subtract radicals with the same index and radicand.
  • Simplify if possible.
  • \sqrt[n]{x}
  • n is index, and x is radicand.
  • Never add radicand and index. Just add the number outside of the radical.

Examples:

\sqrt{3} + 2\sqrt{3} = 3\sqrt{3}

can be added because they have the same index and radicand…

 

4\sqrt[4]{5} - \sqrt{5}

is simplified to as…

4\sqrt[4]{5} - \sqrt{5}

can’t be subtracted because although they have the same radicand, they don’t have the same index..

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WWI Artifact: ‘The Revolution’ at Camp Ruhleben

(Source: http://ww1.canada.com/memory-project – ‘The Revolution’ at Camp Ruhleben, Germany in November, 1918 – Eileen Campbell, Thelma Johnson, Gail Campbell and Joan Chittick)

 

What does this artifact tell us about the First World War?
It tells us that not only the war is happening outside the countries, it’s also happening inside. Because of the declaration of wars, people who were the from the enemy’s country were considered as prisoners, like how British civilians became prisoned in German. Camp Ruhleben is an example, they imprisoned British people as the war started and they were treated horribly. This shows the effect of the war had to the people of their own country.
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Week 9 in PreCalc 11

Something I’ve learned this week is about modeling. No – not the modeling one where you pose for a picture – what I’m talking about is creating a quadratic equation based on a word problem, like regarding revenue, money, finding numbers, projectile motions, etc.

 

There’s actually no need for me to explain what it is as it’s pretty straightforward. All you have to do is analyze the word problem carefully and try and make an equation (of course, with a graph!) based on the problem.

 

Now, let’s take a look at this problem.

 

Find two integers that has a sum of 16, and the greatest possible product.

 

two numbers that has a sum of 16. In other words, it can be written like this algebraically,

x + y = 16

which then, we can arrange into…

y = 16 – x

 

Now, let’s say that

x = 1^{st} number

and…

$latex 16 – x = 2^{nd} number$

 

Now, if we add those two, we should get a sum of 16.

Why? Well, since our first number is x, and we arranged the equation to become y=16-x, which then means that y has the same value as 16-x, then if you add

x and 16-x, you should get 16 as in the first equation you’re adding x and y.

 

Anyways, since we also need to find the greatest possible product, we’re gonna multiply them instead of adding, which we’ll get the equation…

x (16 - x) = y

 

Right now, it’s in the factored form, so it means we already have our x-intercepts or roots, which are:

x = 0 ; and x = 16

Now what? Well, we need to find the value of the vertex, because the x-value is the value of each two integers that we need to find and the y-value is the product that we also need to find.

 

we can find the x-value of the vertex in two ways: graphing and calculating.

 

For calculating, we just need to find the average, which is pretty easy.

0 + 16 = 16

16 / 2 = 8

And x = 0 is your vertex’ x-value.

 

Now, for the graphing, we just need to sketch it. but since we already have the x-value let’s do a sketch…

 

Now, how do we figure out the product? Well, just plug the x-value of vertex in your formula.

 

f(8) = 8 (16-8)

f(8) = 8 (8)

f(8) = 64

 

Well, there you have it!

 

Two integers that has a sum of 16 is 8, and 8 with the greatest possible product of 64!

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Perceptions vs Reality: WWI

Explain the biggest differences between the perceptions of 1914 and the reality of war in World War One.

 

People, especially the poor ones, thought that enlisting to the army would give them a great opportunity to gain wealth, reputation, to show off, etc. Some were even excited to go to the war. However, little did they know that the life they would have was worse than the life they had back to their homes. They thought that after the war, they would live better than before, but the reality was very different. They had only thought that it would last for a year or two, but it lasted double than that. They weren’t prepared, so they didn’t get enough food later especially when the army doubled in size, some got depression due to the environment they live in, some inflicted wounds upon themselves, some committed suicide, and some even went insane. They didn’t get any medical treatment as technology wasn’t very advanced back then, so they got infected and most of them died. Not only did they not get the result they wanted, but it made their lives worse.