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Week 17 in Pre-Calc 11

This week, we have learned trigonometry and I will discuss one of the things I learned: the Sine Law.

 

It’s a very useful to solve any triangle as long as you have:

  • an angle that has a side corresponding it.
  • two angles / two sides / one angle and one side.

Here’s how it works: every angle needs to be written in CAPITAL LETTERS and every side needs to be written in LOWER CASE LETTERS.

angle A corresponds with side a or side BC.

angle B corresponds with side b or side AC.

angle C corresponds with side c or side AB.

 

what’s the formula?

if you’re finding an angle…

\frac{sinA}{a} = \frac{sinB}{b} = \frac{sinC}{c}

 

and if you’re finding a side…

\frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC}

 

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Nuclear Power Debate

As concerns about environment and power usage has grown, usage of nuclear power has become a controversy to many countries. Many debate to use it as continuing to use fossil fuels greatly harms our environment, and many debate not to use it as they are worried about the radiation and economic meltdown in the future as building a nuclear power plant costs billions.

 

My stance on this long debate is anti-nuclear. Firstly, the effects of radiation poisoning from the nuclear accidents, like the one in Chernobyl, was devastating. Other than that, the waste materials they produce emits radiation for thousands of years. And being exposed to radiation isn’t good either as we can see from the effects it brought to countries like Japan that had nuclear accidents before. For example, they couldn’t eat much food because some livestock got radiation poisoning and it can get transmitted to people if they eat them.

Also, although the power plants aren’t “emitting carbon,” the way they get their fuel, or uranium isn’t CO2-free either. In fact, uranium mines inevitably pollute our environment. Wastes they produce are very radioactive and high in temperature, harming our environment.

Furthermore, using nuclear power plants are very costly and can cost until $9 billion, and can take a long time to build. Another disadvantage is that there are finite resources that they can use and they don’t work forever. So they have to make another one again after so many years.

Other than those, having access to nuclear power is very dangerous as there are terrorist groups existing today and if they can get the information, it can get dangerous for the people as they can have nuclear power and make nuclear weapons that they can threaten the world with.

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Week 15 in PreCalc 11

One thing I’ve learned this week is about multiplying rational expressions.

 

It doesn’t really need a lot of explanation as it’s just your simple multiplying of fractions, however with variables this time.

 

Let’s have a recap:

(\frac{2}{5}) (\frac{10}{8}) = \frac{20}{40}

 

We just multiply straight across, but remember we need to simplify our answer.

20/40 = 1/2

 

But we can also do it this way:

It’s like prime factorization method. So, you take out all of the prime factors, then just cancel out. Well, that’s exactly what we’re going to do! But let’s add variables to the fun.

 

So we have this not-so-pretty looking fractions and we might be thinking, how do we solve that?!

Well, I got good news! They’re pretty easy to solve as long as you’re good at factoring!

 

So first, factor everything.

Now that you’ve factored everything, just cancel out same factors, just like what we did earlier in that simple fraction.

 

Next – don’t forget about this! Remember the rule that you cannot have a zero on your denominator? This also applies to this! So, after everything’s factored out, calculate the values of (or other variables) that will result the denominator to equal zero.

 

Now you’re done!

 

If you’re wondering ‘couldn’t we have just cancel numbers out before we factor’? Take note that if numbers have +/- between them, they’re called terms and you cannot cancel them. So we factor them so we can get factors, which has multiplication sign between them.

 

YOU CAN NOT CANCEL TERMS

YOU CAN ONLY CANCEL FACTORS!

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Week 14 in Pre-Calc 11

One thing I’ve learned this week is about reciprocal functions. We’ve always use that word when we divided fractions. We ‘reciprocate’ the fraction we’re dividing with.

Anyways, let’s recall what a reciprocal is:

  • the reciprocal ofis \frac {1}{x}
  • Basically, the reciprocal of a number is just the numerator and denominator switched.
  • Remember that all whole numbers and variables are over one. They have a denominator of one, basically.

 

What we need to remember about this topic are asymptotes, one’s called vertical asymptote (x value), one’s horizontal asymptote (y value).  An asymptote is a line that corresponds to the zeroes of your equation.

 

So, let’s say we have y = \frac{1}{3x + 6}

We must know that we cannot have a zero on the denominator, so we’ll first find out which value of x would result to have a zero denominator.

 

3x + 6 = 0

3x = -6

x = -2

 

Like was stated earlier, asymptotes are the zeroes to the function. So basically, since we just found out the zero of the denominator, we also found out one of the asymptotes. Remember, the vertical asymptote is a value of x, so our vertical asymptote is (x=-2)

Right now, though, we don’t need to bother with the horizontal asymptote. Take note that if the function’s denominator is 1, our horizontal asymptote will always be zero. (y=0)

 

So this is how you graph it:

  • put a dashed line on where x= -2 is.
  • put a dashed line on where y = 0 is.
  • If the slope of your original line (3x + 6) is positive, then you will draw the hyperbolas on quadrant I and III, and if it’s negative draw it on quadrant II and IV.
  • (https://www.varsitytutors.com/hotmath/hotmath_help/topics/quadrants)
  • The hyperbolas must never touch the asymptotes, since they are your non-permissible values.
  • The graph should look like this.:
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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: