Math Practice Problems #2

Previous

Today, I was asked this question:

"Titan and Rhea are the two largest moons of Saturn. The ratio of of the surface area of Titan to the surface area of Rhea is about 34:3. If the diameter of Rhea is about 1530km, what is the diameter of Titan?"

This one's pretty simple. We only need two formulae.

A=4πr²
d=2r

That's for the surface area of a sphere. Now we need to assign values.

A_Titan=4πr_Titan²
A_Rhea=4πr_Rhea²
d_Rhea=1530 km
A_Titan=(34/3)A_Rhea

Now we can start figuring out the diameter of Titan

r_Rhea=1530km/2=765km
4πr_Titan²=(34/3)4π(765²)

Since there's a 4π on each side, we can eliminate it from both sides.

r_Titan²=(34/3)(765²)

Next, we take the square root of both sides

r_Titan=√((34/3)(765²))

Now we multiply both sides by 2.

d_Titan=2·√((34/3)(765²))
↓
d_Titan=5150.74752

Finally, we need to round the answer until the number of digits we end with matches the number we started with, so the final answer is 5150 km.

Physics Practice Problems #1

OK, so someone asked for help with this problem.

Heating 235g of water from 22.6C to 94.4C in a microwave oven requires 7.06 x 10⁴J of energy. If the microwave frequency is 2.88 x 10⁵s^-1, how many quanta are required to supply the 7.06 x 10⁴J?

You only really need from the question are the frequency of the microwaves (2.88•10⁵s^-1), the temperature change (22.6C to 94.4C), and the total energy (7.06•10⁴J). First, we turn the frequency into a wavelength.

λ=c/f
c=299,792,458 m/s
f=2.88•10⁵s^-1
λ=299,792,458/(2.88•10⁵) m•s/s
λ=1040.49603

Next, we need to find the temperature change.
T₀=22.6°C=295.75K
T₁=94.4°C=367.55K

T=T₁-T₀
T=367.55-295.75
T=71.8

Next, we'll need a few constants.
Planck's Constant: h=6.626068•10^-34 m²kg/s
Boltzmann's Constant: 1.3806503•10^-23 m²•kg•s^-2•K^-1
e=2.71828183

Now, we're ready to find the energy of one photon, but we need a formula.

You use all the values in that formula and you get 9.91306903•10^-22 J per photon.

Now all you need to do is divide the total energy by the amount of energy each photon caries.

(7.06•10⁴J)/(9.91306903•10^-22J)
7.12191147•10²⁵

So it takes 7.12191147•10²⁵ photons to heat the water, but we need to remember to round to the appropriate number of digits. We started wit 3-digit numbers, so we have to end with a 3-digit number. The final answer is 7.12•10²⁵ photons.

Thanks to Wikipedia for the picture of the formula.

Energy

There are different forms of energy, like electrical, thermal and chemical, but I'd like to start off with mechanical energy.

Well, what is energy? Energy is the amount of work that gets done. It is the amount of force applied to something for a given distance. There was this saying in my high school physics class: "Work is mad". That means that the amount of work you do (energy you use) on an object is equal to the mass, acceleration and distance multiplied together. I'll give you two formulas right now.

E=mad
E=Fd

The unit for energy is called the joule. It's abbreviated as J and can be written in terms of the unit's we've already learned about.

1J=1Nm
1J=1 m²kg/s²

OK, here's a problem: We have a crate that weighs 500 Newton that needs to be lifted 39 meters to a 13th-story window. How much energy will it take to lift straight up?

F=500kg
d=39m

E=Fd
E=500N·39m
E=19500 N·m
E=19,500 J

So you'd need to use 19,500 joules of energy to get the crate up to the window.

Math Practice Problems #1

While out roaming the Internet, I sometimes encounter people who need help with their homework, so I had the idea of collecting their questions and posting them here.


"If the polynomial 4-7x+2x²-5x³ has degree a, leading coefficient b, and constant term c, then the value of 3a-2b-c is _____."

Well, the highest exponent is 3 from -5x³,and the leading coefficient is always attacher to the highest exponent, so that's -5. The constant is the term that doesn't have an exponent, so that's 4.

So...
a=3
b=-5
c=4

And we can solve the problem.

3a-2b-c
3(3)-2(-5)-4
9-(-10)-4
9+10-4
15

And the answer is 15.

Next

The Periodic Table

There are 118 known elements in the world. Some happen naturally and some are made in a lab. With all those elements, it's easy to get lost if you don't have a good system for sorting them. Fortunately, chemists and the alchemists before them thought of this problem. As they learned more about different elements, they worked out a pretty good way of sorting them called the Periodic Table of Elements.



You can get some very important information just by looking at the Periodic Table, like how many valence electrons it has, how hard it is to make 2 atoms bond, or if they'll even bond at all.

Tangent

Hello, everyone, I'm back from my year-long absence. I promised that I'd write about the tangent (tan) that went with sine and cosine in trigonometry, so here goes. First, we take the same triangle we used for the sine and cosine.



Let's use the same axes, too.



On the first triangle, A, B, and C still represent the sizes of the angles in degrees. The little a, b, and c represent the lengths of each side. Here's how they combine to form tangential measurements:

tanA=a/c
tanC=c/a

Remember the "Toa" part of "SohCahToa"? It stands for tangent equals opposite over adjacent. On the axis drawing, x is adjacent and y is opposite, so

tanθ=y/x

Force #2

Previous

Now, gravitational force, normal force, what are they? What do they have to do with each other?

Well here it is. Gravitational force is the force pulling something down. Normal force… You know how when there’s something on a table, the thing doesn’t sink into the table, but the table doesn’t launch it either? Normal force does that. Normal force is the force an object exerts to counter the force of another object trying to push through it. It’s usually gravity’s equal and opposite reaction. I’ll show you what I mean.

Let’s show up at a beach in southern India, that’s close enough to sea level and the Equator. Let’s bring a 2500 kilogram elephant with us while we’re at it.

2500Kg•9.81m/s²
↓
2500•9.81=24525
↓
24525N

The elephant weighs 24525 Newtons. That being true, it exerts 24525 Newtons of force on the ground just by standing there, and that force acts in a downward direction. The elephant doesn’t sink into the ground, nor does it get launched up into the air, and is pretty happy with that. So, what conclusion can we gather from that? The normal force is equal to the elephant’s weight, but in the opposite direction, pushing up against the elephant’s feet. Yes, gravitational force is 24525N down, so normal force is 24525N up.

Acceleration #5

Previous

Today, we get an artillery on a cliff above the same camp as last time. Osama was spotted through a window seated on the toilet in his outhouse 400 meters away and 100 meters down from our spot. This time, our artillery is better than the one we used last time, firing shells at 100m/s. The question is, how high do we aim it? We’ll need a new formula for that.

0=as_x²/v²•tan²θ+s_xtanθ+ as_x²/v²-s_y

You can see how I derived that for yourself if you want. It uses the same variables as before, so now to plug the numbers in.

0=-9.8•400²/100²•tan²θ+400tanθ+(-9.8•400²/100²-(-100))
↓
0=-9.8•160000/10000•tan²θ+400tanθ+(-9.8•160000/10000+100)
↓
0=-9.8•16tan²θ+400tanθ+(-9.8•16+100)
↓
0=-156.8tan²θ+400tanθ-56.8

Unfortunately, I don’t see it being easily factorable, so we put it through the quadratic formula.

(-b±√(b²-4ac))/2a

a=-156.8
b=400
c=-56.8

Let’s start with the addition version of it.

tanθ=(-400+√(160000-(4•-156.8•-56.8)))/(2•-156.8)
↓
tanθ=(-400+√(160000-35624.96))/-313.6
↓
tanθ=(-400+√124375.04)/-313.6
↓
θ=8.582852416°

Now the subtraction version.

tanθ=(-400-√(160000-(4•-156.8•-56.8)))/(2•-156.8)
↓
tanθ=(-400-√(160000-35624.96))/-313.6
↓
tanθ=(-400-√124375.04)/-313.6
↓
θ=67.38090412°

Well, it would’ve been nice to have one of the answers be obviously wrong, but it doesn’t always work that way. Let’s check both answers.

θ=8.582852416°
↓
tanθ=(-400+√124375.04)/-313.6
↓
0=-156.8tan²θ+400tanθ-56.8

You put the value of tanθ into the equation on the third line of that set, use your calculator, and you get

0=0

…which means θ=8.582852416° is a right answer. Wait, we’re not quite done yet. It’s always a good idea to check the other answer, too.

θ=67.38090412°
↓
tanθ=(-400-√124375.04)/-313.6
↓
0=-156.8tan²θ+400tanθ-56.8

You do the same thing again, putting the answer you got for tanθ into the equation. This time you get

0=0.000000002

Close, but not good enough to be a right answer.

Now we can aim the artillery 8.582852416° above horizontal, fire, and watch Osama’s outhouse get blown up.

Acceleration#5 - Derivation

OK, first I want to bring back 4 equations from last time…

sy=(1/2)at²+v_yt+s₀
s_x=v_xt
v_x=vcosθ
v_y=vsinθ

…and a trigonometric identity.

1/(cos2θ)=tan2θ+1

Now we can derive a new equation to fit what we’ll do today.

s_x=v_xt
↓
t=s_x/v_x
↓
t=s_x/(vcosθ)

sy=(1/2)at²+v_yt+s₀

Since s₀ is the origin, it equals 0 and would not need to be included.

s_y=(1/2)at²+v_yt
↓
0=(1/2)at²+v_yt-s_y
↓
0=(1/2)at²+vsinθt-s_y
↓
0=(1/2)a(s_x/(vcosθ))²+vsinθ(s_x/(vcosθ))-s_y
↓
0=((as_x²)/v²)(1/cos²θ)+s_x(sinθ/cosθ)-s_y
↓
0=(as_x²/v²)(tan²θ+1)+s_xtanθ-s_y
↓
0=as_x²/v²•tan²θ+s_xtanθ+ as_x²/v²-s_y

Force

A force is a push or pull on an object. It can be what you use to push a box across the floor, the gravity that pulls things down when they fall, or what keeps a maglev train floating above the track. An object can either use force directly like you do when you push the box, or as a field force like gravity and magnets. The difference is that the object that’s pushing has to touch the object that’s being pushed, but the field force acts from a distance. You’re probably expecting a formula, well here it is:

F=ma

…And here’s what it means: To find the force acting on an object, you multiply the mass of the object times its acceleration. That means force has a magnitude (size) and a direction.

Yes, there’s a special unit for force, too. If you multiply the units for mass and acceleration, you get Kg•m/s² and

1Kg•m/s²=1N

If you have a 1 kilogram object and it’s accelerating at 1m/s², a force of 1 Newton is acting on it.

But there’s always more than one force acting on an object. Say you were making a 1 Kg weight accelerate upward at 1m/s². Here’s a helpful number:

9.81m/s²

That’s gravitational acceleration near sea level on Earth. The Earth’s gravity is pulling every kilogram down with a force of 9.81N. You’re actually pushing the weight up with a force of 10.81N. Both those forces together make a net force of 1N upward, so the 1 Kg weight can accelerate up at 1m/s².

Next

Mass

Mass is one of the basic ways of measuring things in science. Chemistry measures things in grams, physics likes using kilograms.

grams = g
kilograms = Kg

Many people use mass and weight to mean the same thing, but they are different. Weight is actually a force and can change while mass stays the same.