Math Notes
Sample Math Solutions
Linear
Equations Standard Form and Perpendicular:
Determine the equation in
standard form of the line that passes through (5,5) and is
perpendicular to the line 5x + 2y =10?
Solution:
If two
lines are perpendicular their slopes are negative reciprocals. First
solve for y to get the slope of your line: 5x + 2y = 10 --> 2y =
-5x + 10 --> y = -(5/2)x + 5 The slope of the new line is going
to be 2/5 and it goes through (5, 5). We have y = (2/5)x + b
Now
plug in your point (5, 5)
5 = (2/5)5 + b --> b = 3
So the
equation of your new line is going to be:
y = (2/5)x + 3
now
standard form:
-(2/5)x + y = 3
2x - 5y = -15 I multiplied both
sides by -5
Systems of Equations:
How many
minutes after 3:00 will the hands of a clock
overlap?
Solution:
I prefer to look at it as
a position / velocity question. It's simpler and more elegant that
way. That is let the minute hands position be x = 0 and velocity be
60. Let the hour hands position be 15 and velocity 5. Why? It goes 5
minutes every hour and the minute hand goes 60 minutes every hour.
The units are arbitrary and we can imagine the clock laid out on a
number line. Were do the two intersect? Now it's an easy problem.
Let t = number of hours
Position minute hand:
0 +
60(t)
Position hour hand:
15 + 5(t)
now when they intersect
the positions are the same:
0 + 60t = 15 + 5t
55t = 15
t =
15 / 55 = 3 / 11 = .2727... hours or 16.3636...
minutes.
Physics Notes
Sample Physics
Solutions
Kilowatt Hour Problems:
A
nuclear power plant has an electrical power output of 1500 Megawatts
and operates with an efficiency of 35%. If excess energy is carried
away from the plant by a river with a flow rate of 106,000 kg/s,
what is the rise in temperature of the flowing
water?
Solution:
This is an interesting problem. First,
how much water is 106,000 Kg/s? Since on cubic meter weighs 1,000 Kg
it would be 106 cubic meters / second. That might be say a river 50
m across and 2 meters deep with a flow velocity of about 1 meter per
second. We are also assuming all the heat not used to power the
plant goes into the water. This is an incorrect assumption. So now
for the solution:
.65 * 1500 MW = 975 MW (HEAT)
1 Megawatt =
10^6 Watts = 10^6 J/s
975 x 10^6 Joules per Second Heat
Now
find the Joules imparted into each KG of Water:
(9.75 x 10^8 J/s)
/ (1.06 x 10^5 Kg/S) =>
9.20 x 10^3 J/Kg
Now get the
specific heat of water:
4186 Joules required to heat 1 Kg 1
degree K
Now (9.20 x 10^3 J/Kg ) / 4190 J/Kg K
2.20 K
Hotter
To extend this research how much of the heat is dissipated
before heating the river.
An LED light uses 6
Watts and has the same lighting capacity as an incandescent that
uses 60 Watts. Compare energy costs after 30,000 hours. Assume .10
per KWH
Solution:
First, lets look at the units. A KWH or
Kilowatt hour is 1,000 Watts for one hour. A Watt is 1 Joule / 1
second. Therefore, a Kilowatt hour is 1,000 Watts x 3,600 S or
3,600,000 Joules!
Incandescent
30,000 hrs x
60 Watts = 1,800,000 Watt Hours or 1,800 KWH. 1,800 KWH x .10 /KWH =
$180
dollars!
LED
30,000 hrs x 6 Watts = 180,000 Watt Hours or 180
KWH. 180 KWH x .10 /KWH = $18 dollars
For the difference subtract
$180 - $18 = $162. $162 represents the energy savings of using LED
lights. Of course, you also save on all those incandescent
replacements.
Pressure Problems:
A
container is filled to a depth of 20.0 cm with water. On top of
water floats a 30.0 cm thick layer of oil with a specific gravity of
0.700. What is the absolute pressure at the bottom of the
container?
Solution:
First lets
just get the equivalent of water in meters. The oil is .7 x .3m =
.21m or equivalent to .21
meters of water. So the total is:
.2 m + .21 m = .41 m.
Now
we need the pressure at the bottom of a .41 meter column of water.
Now 1 Pascal = 1 Newton / m^2 So lets get the Newtons at the bottom
of cubic meter of water. You could memorize it but I would figure
it. A cubic meter of water weighs 1,000 Kg. F =
mg =
9.8 x
1,000 =
9,800 Newtons and it's 1 meter squared at the bottom. So a 1 meter column
of water produces 9,800 Pascals pressure. .41 x 9,800 = 4,018 round
to 4,000 Pa. Now finally don't forget the atmospheric pressure
equivalent to about 100,000 PA. So the total is 104,000
PA.
Shed some light on your homework with CREE LED Powered Bulbs LED
lights can provide better lighting, motivate your studies, save
energy, reduce C02, No UV, save on bills (ultimately save money),
and not poison you with toxic mercury.
Contact us:
email: berrtus@gmail.com