Exponent of Exponents

                                                                         EXPONENTS

When we calculate the exponent of some number that is already written as an exponent we have to multiply these two exponents in order to get the result. (I know that this explanation isn't very thorough, and is sort of confusing, so I've managed to make up an example to make it sound easier: 

1) (2^3)^2 = 2^3x2 = 2^6

                                                                               OR

2) (5^4) = 5^4x5 = 5^20

If instead of a real number we write an x, which is a variable, we will get the general rule that can be written as:

(x^m)^n = x^mxn

WARNING: Be careful, there is a BIG difference between example number 1 and the following example:


2^3 x 2^2 = 2^3+2 = 2^5 (INCORRECT)

I hope you learned something new!

Simplifying 24^2 / 2^2

Ratio Between Gymnasium & Classroom

The Dimension of the school is 32m x 25.8m x 7m = Vg = 5,779.2 m^3The volume of one of the classrooms = Vc = 104.16 m^3
How many times would the classroom fit inside the gymnasium?
The answer is Vg / Vc = 55.48 times

Independant Study Proposal

Canada, Germany, New Zealand, and Japan

For this homework assignement, I decided to do New Zealand as the fourth country as well because I started doing it in class, eventhought I was not supposed to. Below I have calculated for Canada, Germany, New Zealand, and Japan.

Canada: We are looking for the Death rate!
P(2012) = P(2011) + P(2011) (B - D)
34108860 = 34030590 +  34030590  (0.01028 - D)
- 34030590
78270 = 34030590 (0.01028 - D) Use distributive property here!
78270 = 349834.5 - 34030590D
-349834.5
-271564.5    =  -34030590D
-34030590   = -34030590
Since both the denominator and numerator are negative, we can transform them to be positive, which will give us the same result because a negative divided by a negative is positive, and positive divided by positive is positive.
D = 0.00798

Germany: We are looking for the population in 2012!
P(2012) = P(2011) + P(2011) (B - D)
P(2012) = 81471830 + 81471880 (0.0083 - 0.01097)
Since we have both Birth and Death rate, we can immediately subtract them, and move on to the distributive property.
P(2012) = 81471830 - 217529.8
P(2012) = 81254300

New Zealand: We are looking for the birth rate!
P(2012) = P(2011) + P(2011) (B - D)
4045249 = 4027947 + 4027947 (B - 0.00715)
-4027947  
17302 = 4027947B
+28799.8
46101.8  =   4027947B      
4027947 =   4027947   
B = 0.01144

Japan: We are looking for the death rate!
P(2012) = P(2011) + P(2011) (B - D)
126124097 = 126475700 + 126475700 (0.00731 - D)
126124097 = 126475700 + 924537.4 - 126475700D
126124097 = 127400237.4 - 126475700D
-127400237.4
-1276140.4 = -126475700D
-126475700= -126475700
D = 0.0101

Total World Population

From the graph above, we can see a huge difference between the number of people in the developed and in the developing countries. As we can see starting from the 1930s, the number of people in both regions start to increase, but the slope in the developing region is much steeper. This leads to a bigger increase in the population of the developing region than in the developed one. This is because we have much bigger birth rates than death rates in the developing countries. But we can also see that as we get closer to 2100, the populatin in both countries are not increasing anymore. (Will the planet become too small for all of us? I'm starting to get worried.

Births, Deaths, and Algebra

                                                 

                              Births, Deaths, and Algebra

 Something you need to know:

19.5 births/1000 (1000= be careful! Not /100)
P=Population
P(2012) = P(2011) + P(2011) (B-D)
P(2012)= Want to estimate population for 2012
P(2011) = This is the population we started within 2011
P(2011) (B-D) = This is the growth for 2011
B = Birth rate in decimal form
D = Death rate in decimal form

Macedonia:
Birth Rate: 11.9/1000 = 0.0119
Death Rate: 8.9/1000 = 0.0089
The Equation:
P(2012) = P(2011) + P(2011) (B-D)
P(2012) = 2,114,550 + 2,114,550 (0.0119 - 0.0089)
0.0119 - 0.0089 = 0.003
2,114,550 + 2,114,550 times 0.003
Here, I made a mistake, and my solution at the end was wrong. I first added 2,114,550 + 2,114,550 and then multiplied 0.003! Everybody should watch out at this stage. What I then figured out is that I have to use PEMDAS (Parantheses, Exponents, Multiplication, Divisoin, Addition, Subtraction) in order! This is called the order of operations!
The final result is: 2,120,893.65
To simplify this, the final, FINAL result is 2,120,894. This is the population for the people in 2012.

Future Macedonia:

What will be the population in Macedonia in 2013, 2014, 2015, and 2016? How will we find this out?

Birth Rate: For calculating the average birth rate, I've used values from 2005 - 2011 and I have calculated that the average birth rates are decreased by 0.022 every year!
I got that estimated values of the birth rate is as following:

2011:	2012:	2013:	2014:	2015:	2016:
11.87	11.85	11.63	11.61	11.58	11.56

Death Rate: For calculating the average death Rate I did the same thing, but including different numbers:

2011:	2012:	2013:	2014:	2015:	2016:
8.90	8.928	8.956	8.984	9.012	9.04

I found that the average increase from year to year will be 0.028

Finally I chose to use these values to calculate the estimated values for the Macedonian population:

P(2013) = 2,120,894 + 2,120,894 (11.848 - 8.928 / 1,000)
= 2,120,894 + 2,120,894 times 0.00292
= 2,120,894 + 6193.27 = 2,127,087

P(2014) = 2,127,087 + 2,127,087 (11.61-8.984/1000)
= 2,127,087 + 2,127,087 times 0.002626
= 2,127,087 + 5585.7 = 2,132,672

P(2015) = 2,132,672 + 2,132,672 (11.58 - 9.012 / 1,000)
= 2,132,672 + 2,132,672 times 0.002568
= 2,132,672 + 5,47.7 = 2,138,148

P(2016) = 2,138,148 + 2,138,148 (11.56 – 9.04 / 1,000)
=  2,138,148 + 2,138,148 times 0.00252
= 2,138,148 + 5388 = 2,143,536
Now that I know the formula for each year, I can find out what the population for Macedonia will be even in 2050! It would just be a little tedious though, or I can use a graph to do my estimation easily.