Adding Vectors: The “chart thing”

Especially when tutoring physics 12, vector addition comes up.  Let’s use the “chart thing”:

I took Physics 12 in 1988.  In that class, we added vectors component-wise.  However, at university they used sine law and cosine law.  I haven’t seen someone use component-wise addition in over twenty years – until this week.  Now, they call it “the chart thing:)”

Let’s imagine we’re adding the following two vectors:vector 1vector 2

Well, first we use trig to resolve each into horizontal (x) and vertical (y) components:

vector 1 resolvedvector 2 resolved

Now, looking above, we see that for the first vector, the x component is pointing left:  it’s negative.  A downward component would be negative as well, but we don’t have any downward components in this problem.

Now, to the chart:

vector x component y component
vector 1 -12.3 8.60
vector 2 12.0 18.5
total: result vector -0.3 27.1

You just add downwards to get the total x and y components of the resultant vector.  We see that the result has -0.3 for its x component and 27.1 for its y component.  We can build it as follows:

resultant vector
Now, use a² + b² = c² to get the hypotenuse. Of course, you’ll still get 27 (two sig digits), since 0.3 is too small to have any significant effect on the vertical component of 27.1. Use 2ndtan(27.1/0.3) – you may also know it as arctan(27.1/0.3) – to get the angle of 89°.
resultant vector solved

Since there were only two significant digits in the original problem, we should give the answer with two.  (This isn’t always true, technically, but it’s a good rule of thumb.)  Hence, we give the answer in two sig digs: 27 m/s 89° N of W.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

 

Inequalities: Phone Plans

Which phone plan to choose?  A little math tutoring can help you decide.

Let’s assume we have two choices:

Plan A $12 per month $0.10 per text
Plan B $18 per month $0.07 per text

How do you decide which plan to use?

Let’s say you’re buying the plan for a month.  Let x be the number of texts you send.  Then your cost is

12 +0.10x for Plan A

18 +0.07x for Plan B

You pay more up front with Plan B, but its texts are cheaper.  There will be a specific number of texts at which the plans cost the same:

(1)   \begin{equation*}12 + 0.10x =18 + 0.07x\end{equation*}

Using algebra, we solve for x:

-0.07x          -0.07x

(2)   \begin{equation*}12 + 0.03x= 18\end{equation*}

-12             -12

(3)   \begin{equation*}0.03x=6\end{equation*}

(4)   \begin{equation*}\frac{0.03x}{0.03}=\frac{6}{0.03}\end{equation*}

(5)   \begin{equation*}x=200\end{equation*}

At 200 texts, the two plans are equal in price for one month.  Any more texts, and Plan B must be cheaper:  it’s less per text.

Now, let’s explore the situation from the point of view of inequalities.  We ask, “For what number of texts will Plan B be cheaper?”

(6)   \begin{equation*}18 + 0.07x \le 12 + 0.10x\end{equation*}

 

-18    -0.10x          -18      -0.10x

(7)   \begin{equation*}-0.03x \le -6\end{equation*}

(8)   \begin{equation*}\frac{-0.03}{-0.03}\le\frac{-6}{-0.03}\end{equation*}

Note that algebraic operations are the same for inequalities as for equations, with one important exception:  you must flip the sign if you divide or multiply by a negative number.  (Not by positive, just by negative).

In our case, we are dividing by -0.03, so we must flip the sign:

(9)   \begin{equation*}x\ge 200\end{equation*}

The two methods make the same conclusion, but from different points of view.  The equation tells us that, at 200 texts, the two plans cost the same.  The inequality, on the other hand, tells us that if we make more than 200 texts, we’ll pay less with Plan B.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Dependent Variable, Independent Variable, Test and Control

When tutoring Biology 12, terms involved in the scientific method need definition.  Let’s look at some.

In an experiment, there is one variable whose value you must wait to see, and (at least) one that you purposely set.  The variable you set is called the independent variable.  The variable you’re curious about is the dependent variable.

A control group – or simply “control” – is one that you leave alone.  You impose changes on the test group by changing the value of the independent variable.  Then, you give the changes time to affect the test group.  Finally, you measure the test group’s dependent variable in order to discover the effect of the changes you imposed.

The typical example to illustrate the roles of these variables is a plant-growing experiment.  If you want to know if using fertilizer accelerates plant growth, your control group will be plants that don’t get fertilizer.  The test group will be plants that do get fertilizer.  The independent variable will be how much fertilizer you give to each plant.  Finally, the dependent variable will be how much each plant grows over the course of the experiment.

There are other terms that relate to the scientific method, but those are generally better documented.  However, we might look at them here in a future post.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC

Neutrons and Isotopes

Tutoring high school chemistry, isotopes are an early topic.  Let’s have a look….

From basic atomic structure, we know that an atom’s mass comes from its number of protons plus its number of neutrons.  Take fluorine, for example:  its mass is 19.  It’s got 9 protons (because its atomic number is 9), so it must have ten neutrons.

But what about chlorine, whose mass is listed at 35.5?  (Possibly, it says 35.453 or something like that, depending on which periodic table you’re using.  By the way:  you can always tell the mass because it’s the one that can have a decimal, whereas the atomic number is always a whole number.)  If chlorine’s mass comes from its protons plus its neutrons, how can its mass be 35.5?  You can only have whole numbers of protons and neutrons.

The answer comes from the concept of an isotope.  Isotopes are like different versions of the same type of atom.  The number of protons is what defines the type of atom you have.  We see chlorine’s atomic number is 17, so it must always have 17 protons.  However, its number of neutrons can vary:  about 75% of chlorine atoms have 18 neutrons, while the other 25% have twenty.  Therefore, 3 out of 4 chlorine atoms (which is 75%) have a mass of 35, while the other 1 out of 4 has a mass of 37.  Let’s take the average:

(1)   \begin{equation*}mass_a_v_e=\frac{35+35+35+37}{4}\end{equation*}

Leading to:

(2)   \begin{equation*}mass_a_v_e=35.5\end{equation*}

So that’s how you get a mass of 35.5.  No chlorine atom actually has a mass of 35.5; some have a mass of 35, while others have mass 37.  More have mass 35, bringing the average to 35.5.

When you’re calculating the number of neutrons, you should round the mass to the nearest whole number, then subtract the atomic number.  For example, to calculate the neutrons in boron (atomic number 5, mass 10.82), we would round the mass to 11 and subtract 5, giving 6 neutrons.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC

Metric System: decimal places and -re

When you tutor sciences, you’re immersed in the metric system.  Let’s talk a bit about it.

The metric system uses multiples of 10.  Therefore, to convert from any metric unit to any other one, you just need to move the decimal point;  you needn’t do any math.   However, you do need to know the metric prefixes, so you know how many places to move the decimal:

kilo
k
hecto
h
deca
da
metre (m)
gram (g)
litre (L)
deci
d
centi
c
milli
m
0.001 0.01 0.1 1 10 100 1000

From the table above, you can see that 1 metre = 0.001 kilometres.  Or, using abbreviations, 1 m = 0.001 km.

Note that the prefixes have their own abbreviations, and the base units (metre, gram and litre) have their own.  mm can’t be confusing because the m in front is the abbreviation for the prefix milli, whereas m in the back is for metre.  Similarly, dL would mean decilitre.

People often ask about the capital L for litre.  It’s true that except for L (and a few other rare instances we won’t get into), you should never use uppercase letters for the metric system.  The reason for the exception with L is that a lowercase l can look like a 1.  To avoid the possible misunderstanding, “l” for litre is either written capital or else in handwriting.

Now, to convert from one unit to the other:  the table tells all.  They key is knowing where the decimal is now, and how many places you’ll have to move it.  For instance, imagine you have 32 cm and you want hm (hectometres). Looking at the chart, you see that from centi to hecto is four jumps left.  (Don’t count centi; count the jumps to hecto).  Therefore, we must move the decimal four jumps left as well.  (In 32, the decimal is of course at the end, since it’s not written.)

32 cm=0.0032 hm

Another example:  how many cm is 2.3m?  Well, looking at the chart, we see centi is two jumps right from metre.  So, move the decimal two jumps right:

2.3m = 230cm

You knew that anyway, since of course there are 100 cm in a metre :)

One more thing:  in metric, it’s -re, not -er.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC

 

Chlamydia and Gonorrhea: infertility risks

Most people have probably heard of chlamydia and gonorrhea.  They are both bacterial STDs.  Since neither is life-threatening (as far as I’ve ever heard), they don’t get as much attention as AIDS or syphilis.

A risk with both chlamydia and gonorrhea is that they can be asymptomatic – i.e., the victim doesn’t have any symptoms, so doesn’t know they’ve caught the disease.  However, both chlamydia and gonorrhea, operating undetected, can cause scarring of the reproductive tract.  In males, the vas deferens (aka ductus deferens) can become scarred over; in females, the same can happen to the oviducts.  The individual can thus be rendered infertile.

Both chlamydia and gonorrhea, being bacterial, are treatable with antibiotics.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Homeostasis, negative feedback, and thyroxin

Tutoring biology, the concept of negative feedback is important to explain.  Negative feedback is used by all organisms to maintain their living state.  You also use it while driving your car.

More or less, the body needs all its processes to proceed at a constant, continuous rate.  Change is the enemy for biological entities.  However, as you interact with your surroundings – eating, exercising, and so on – change is imposed on your internal environment.  Negative feedback means that when a change happens, your body responds to negate that change – i.e., to bring itself back to normal.  Once back to normal, your body stops its negative response.  It won’t respond again until another change brings it out of normal range.

Consider driving a car – specifically, steering.  Imagine the road is straight, but your car starts drifting to the right.  You correct by steering left, to bring the car back on course.  In so doing, you are applying negative feedback.  The car is going too far right, so you oppose that change of course by steering left.  Once the car is back on course, you stop correcting:  you let the wheel slide back to center.  You won’t steer again until the car begins to drift off course once more.  That process of opposite, corrective response is negative feedback.  You can also call it a feedback loop, since you base your reaction on what is already happening.

Biology has many examples of negative feedback, but we’ll look at the one involving the hormone thyroxin:

The hypothalamus – in the brain – monitors the body’s metabolism.  If the metabolism gets too low, the hypothalamus stimulates the anterior pituitary.  In response, the anterior pituitary releases a hormone that causes the thyroid to release thyroxin.  Thyroxin increases the body’s metabolism.

Once the metabolism reaches an acceptable level, the hypothalamus stops stimulating the anterior pituitary.  In turn, the anterior pituitary ceases to release the thyroid-stimulating-hormone.  Therefore, the thyroid stops releasing thyroxin.  The body’s metabolism remains constant – until another external change depresses it again.

Therefore, the connection among the hypothalamus, anterior pituitary, and thyroid gland constitutes a negative feedback system that maintains the body’s metabolism at a constant rate.  This negative feedback system promotes homeostasis.  Homeostasis means the maintenance of a constant internal environment.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Cell efficiency

When you tutor Biology 12, one topic that comes up is cell efficiency.  It’s a bit tricky for some people, because it involves some math.

Putting it simply, imagine a cell is a sphere.  Its volume is what it needs to maintain, whereas its surface area is where it gets its supplies.  You can quickly realize that it’s best to have a big surface area compared to volume (or surface area to volume ratio), so the cell can easily get enough supplies to feed its volume.  Efficiency, in this context, refers to the cell’s surface area to volume ratio:

(1)   \begin{equation*}Eff.=\frac{SA}{V}\end{equation*}

As the radius of a cell grows, its surface area grows, but its volume grows more quickly.  Therefore, its surface area to volume ratio decreases:  its efficiency decreases.

Therefore, cells are better off being small – which is why most cannot be seen with the naked eye.  Ultimately, this same principle (of efficiency) is why a rat can run up the side of a building with ease, but a human cannot.  The human’s muscles, since they have such greater volume than the rat’s, are much less efficient.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC

Conservation of momentum

Every year, the physics tutor fields a few questions on conservation of momentum.  It’s an interesting phenomenon because you can use it to explain some familiar, everyday situations.

Momentum is mass times velocity.  Something that is 50 kg, traveling at 12 m/s, has a momentum of 600 kgm/s.  It’s a vector, so two momentums can cancel each other out if they have opposite directions.

One great example of conservation of momentum is how a jet boat works.   The motor takes water, which has an initial momentum of zero, and pushes the water, giving it velocity.  The momentum the water gains needs to be canceled somehow, since total momentum must remain constant.  That’s why the boat goes forward:  to cancel out the backward momentum the water has been given.  The boat gains the same momentum forward that the water gains backward.  From the point of view of physics, that’s why a jet boat moves forward.

Thanks for dropping by, and come again!

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Does 0.33333….. really equal 1/3?

Hello.  What rain yesterday, here in Campbell River!  Well, we sure needed it.  It’s nice to have more seasonal temperatures after the oven that was last week.

A math tutor often encounters the topic of converting decimals to fractions.  Terminating decimals are easy:  for instance, 0.9 is 9/10.  Then, 0.31 is just 31/100.  As well, 0.222 is 222/1000, which reduces to 111/500.

What about repeating decimals, such as 0.333333…….?

Well, there’s an algebraic trick for that:

Let x=0.3333…..(Note that x=1x; we just don’t usually write the one.)

Then

(1)   \begin{equation*}10x=3.33333......\end{equation*}

(2)   \begin{equation*}             1x =0.33333......\end{equation*}

Subtracting (2) from (1) gives

(3)   \begin{equation*} 9x=3.00000......\end{equation*}

Of course, 3.00000…. = 3, so

(4)   \begin{equation*}9x=3\end{equation*}

Next, divide both sides by 9 to isolate x:

(5)   \begin{equation*}\frac{9x}{9}=\frac{3}{9}\end{equation*}

Finally, reducing gives

(6)   \begin{equation*}x=\frac{1}{3}\end{equation*}

Recall, we began by defining x as 0.333333…..Now, since we see x is also equal to 1/3, we know that it must be true:

(7)   \begin{equation*}0.33333......=\frac{1}{3}\end{equation*}

Have a great day, and come back for more hints.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.