Problem set #4

Problem 

# 13

 In this diagram, the smaller circles touch the larger circle and touch each other at the center of the larger circle. The radius of the larger circle is 6. What is the area of the shaded region? (look at page to see diagram)

Solution

First you find the area of the big circle. The formula to do so is πr². 

This would appear like π6² = 36π 

(You would put 36π instead of the expanded form because it's exact)

Now you find the area to one of the smaller circles inscribed in the the larger circle. 

You use the same formula πr² which results in π3² 

(We use 3 because the radius is half of the larger circles radius) 

π3² = 9π(2) = 18π

(You would multiply 9π by two because there are two circles inscribed in the larger circle)

The final step is to subtract 18π from 36π. This would look like 36π - 18π = 18π

The area of the shaded area is 18π

Why do you like this question? 

I like this question because it involves circles and I enjoy finding they're area. It was as well very simple to find out and it gives me more practice using exact answers. 

What have you learned about the process of problem solving? 

I've learned that I shouldn't round my answer till the last step. I sometimes do this without noticing but I'm breaking out of this habit. 

Who Wants to Be a Mathematician Overview

10 new things I've learned about mathematics
1.A squared + B squared= C squared

    -This formula is only true on a plain , not at North Pole.

2.There are an infinitive number of fractions between 0 and 1 
3.A cubed + B cubed = C cubed does not exist.
4. Carl Gauss was a famous mathematician who was said to amaze his teacher by summing all integers from 1 to 100. 
5.The Pythagorean Theorem was most likely not discovered by Pythagoras but by an Asian mathematician 500 years earlier.  
6. x to the power of 4 + y to the power of 4 + z to the power of 4 + t to the power of 4 = t to the power of 4 has been discovered ( person who discovered was awarded )
7.There is a nonstop number of Pythagorean  triplets.
8. Math is all about making mistakes.
9. Questions aren't as complicated as you perceive them to be.
10."Alpha" usually represents an angle.


What I learned from "The Talk"
I learned most of the points stated above and that I'd rather be a mathematician then a millionaire. By being a mathematician you can earn the prizes (there are 7 of them in total) that the America Mathematics institute provides.  By solving each one of these equations you earn a million dollars in reward so you can end up being a billionaire instead of a millionaire. You're as well guaranteed a job in what your good at most.


How did "The Workshop" help me in future math learning?
This workshop was hosted by university students. It was interesting to see how they would explain questions and solve them knowing that in a couple of years, you could be one of them. I learned to not round Pi because when doing that, you don't end up with an exact answer. In a funny way, math can be fun when your understand all the steps your taking and why. Even though at times it's frustrating , your always satisfied with that sense of completion at the end of each question. 

Problem set #3

Question 14:
Carly takes 3 steps to walk the same distance as Jim walks in 4 steps. Each of Carly's steps covers 0.5 meters. How many meters does Jim travel in 24 steps?

Answer:
Carly's steps cover 0.5 meters and when she walks 3 steps that equals the distance Jim covers in 3 of his own steps.
So 0.5 meters x 3 of Carly's steps = 1.5 meters.
1.5 meters = 4 of Jim's steps.
Now you divide 24 steps by 4 steps =  6 steps
Now this equals 6 steps mulitplied by 1.5 meters giving you the answer of 9 meters.
John walks travels 9 meters in 24 steps.

Why do I like this question?
I like this question because the first time I did this I thought it was extremly easy but ended up with the wrong answer. It gave me determination. 

What have I learned about problem solving?
I've learned that I need to double check my answers and make sure my calculations are correct. When I first did this I read it as :
"Carly takes 4 steps to walk the same distance as Jim walks in 3 steps. Each of Carly's steps covers 0.5 meters. How many meters does Jim travel in 24 steps?"
 I messed up the integers in the process making my reckoning wrong. I had to do everything over again to get the correct answer. This wouldn't have occurred if I hadn't rushed through it and double checked. No matter how easy an equation looks you shouldn't skip steps because with one little mistake, it messes up your whole calculation.