Upcoming SlideShare
×

# Project Ko!

527 views

Published on

Published in: Sports, Technology
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total views
527
On SlideShare
0
From Embeds
0
Number of Embeds
38
Actions
Shares
0
6
0
Likes
0
Embeds 0
No embeds

No notes for slide

### Project Ko!

1. 1. Problem no.1 Trig Identities
2. 2. Can u help me to prove this trig identity? <ul><li>In solving these problems, you have to work on the complicated side first. But in this case both of them are complicated so we will solve them together. </li></ul>
3. 3. <ul><li>Firstly, you have to put a boundary </li></ul><ul><li>between them or what we called our </li></ul><ul><li>“ Red Great Wall of China”. </li></ul><ul><li>Left side : </li></ul><ul><li>We recognized that sin2x is a double </li></ul><ul><li>angle identity and so as the cos2x </li></ul><ul><li>Right side : </li></ul><ul><li>cot²x+1=csc²x </li></ul>
4. 4. <ul><li>Left side: </li></ul><ul><li>I will write the long term for double </li></ul><ul><li>angle identities to make things clear </li></ul><ul><li>for you. </li></ul><ul><li>Right side: </li></ul><ul><li>We’ll leave cotx the way it is </li></ul><ul><li>(there’s a reason for that) </li></ul><ul><li>Then, sec²x=1/cos²x </li></ul><ul><li>csc²x=1/sin²x </li></ul>
5. 5. <ul><li>Left side: </li></ul><ul><li>(Numerator) We can add sinxcosx+cosxsinx since they are same terms.. </li></ul><ul><li>(Denominator) </li></ul><ul><li>cosx · cosx= cos²x </li></ul><ul><li>sinx · sinx= sin²x </li></ul><ul><li>So it will equal to 1+cos²x-sin²x </li></ul><ul><li>We could recognize that </li></ul><ul><li>1-sin²x=cos²x </li></ul><ul><li>Right side: </li></ul><ul><li>We can already change cotx=cosx/sinx </li></ul><ul><li>The equation is divided by 1/sin²x, we can take the reciprocal of it and multiply it to the equation. </li></ul>
6. 6. <ul><li>Left side: </li></ul><ul><li>2 is reduced and cos as well. We’ll </li></ul><ul><li>get sinx/cosx </li></ul><ul><li>Right side: </li></ul><ul><li>so the equation is reduced to </li></ul><ul><li>sinx/cosx </li></ul><ul><li>Eventhough they are already the </li></ul><ul><li>same we can still reduce it… </li></ul>
7. 7. <ul><li>Sinx/cosx = tanx </li></ul><ul><li>Thanks for your help..hehe.. </li></ul>
8. 8. Problem no. 2 Permutations
9. 9. I’m J.r. Manleigh and I am the manager of the new band called “Private”. I have here the list of the possible provinces that we can have the concert. Pangasinan Bataan Bukidnon Leyte Batangas Sulu Romblon La Union Surigao Del Norte Bohol Cavite Surigao Del Sur Cebu Laguna Mindanao(green) Visayas(yellow) Luzon(red)
10. 11. If the band plays 6 provinces in total, how many different concert schedules can be created if there is no restrictions in which they play? <ul><li>Order is important thing in a touring schedule, so use permutations. </li></ul><ul><li>Out of 12 possible provinces, the band wants to create a schedule with </li></ul><ul><li>only 6. </li></ul><ul><li>Pick Formula : </li></ul><ul><li>n P r = n ! </li></ul><ul><li>( n - r )! </li></ul><ul><li>nPr is read as “n Pick r” </li></ul><ul><li>where n is the number of objects to pick from </li></ul><ul><li>r is the no. of objects to be arranged </li></ul>
11. 12. <ul><li>n P r = n ! </li></ul><ul><li>( n - r )! </li></ul><ul><li>14 P 6 = 14 ! </li></ul><ul><li>( 14 - 6 )! </li></ul><ul><li>14 ! </li></ul><ul><li>8! </li></ul><ul><li>= 2,162,160 </li></ul><ul><li>or using your calculator… </li></ul><ul><li>14 P 6 = 2,162,160 different concert schedules </li></ul>
12. 13. If the band the band must play all provinces in Luzon, 3 in Visayas and 2 in Mindanao how many different schedules can be created? <ul><li>The 6 provinces in Luzon can be ordered in 6 P 6 ways. </li></ul><ul><li>The 3 provinces in Visayas can be ordered in 3 P 4 ways. </li></ul><ul><li>The 2 provinces in Mindanao can be ordered in 2 P 4 ways. </li></ul><ul><li>6P 6 · 4P 3 · 4P 2 = 207,360 different schedules </li></ul>
13. 14. If the band decides not to go to Bataan and Sulu (because many rumors about those provinces that they are anti-Private people). How many different schedules can be created if the band plays all provinces in Luzon and Mindanao , and only 3 for Visayas ? <ul><li>We can already ignore Bataan and Sulu from the selection pool. </li></ul><ul><li>The first 5 cities in Luzon can now be ordered in 5P5 ways. </li></ul><ul><li>The 3 cities in Visayas can be ordered in 4P3 ways. </li></ul><ul><li>The 3 cities in Mindanao can be ordered in 3P3 ways. </li></ul><ul><li>So… </li></ul><ul><li>5P5 · 4P3 · 3P3 = 17280 ways that the concert schedule can be created </li></ul>
14. 15. Problem no. 3 <ul><li>Logarithms </li></ul>
15. 16. Rio really likes the model of Honda Civic 2007 which will cost you a brand new one \$24, 950 but cannot afford to buy it now. She’s planning to save up some money first then buy it. After 3 years, she will have enough money to buy a used car Honda Civic 2007. But she has 3 dealers to choose from if she wants to get a low priced car. <ul><li>Andrei who offers a Civic that depreciates 0.02% daily for 3 years. </li></ul><ul><li>Danny who offers a Civic that depreciates 0.5% monthly for 3 years. </li></ul><ul><li>Ronnie who offers a Civic that depreciates for 0.15% weekly for 3 years. </li></ul><ul><li>Who will be the best to go to if she wants the cheapest one? </li></ul>
16. 17. <ul><li>Andrei who offers a Civic that depreciates 0.02% daily for 3 years. </li></ul><ul><li>A o = 24, 950 </li></ul><ul><li>model = 0.02% -> 0.0002 (percent to decimal) </li></ul><ul><li>t = 365 x 3(because 365 days in a year and it is for 3 years) </li></ul><ul><li>A = ? </li></ul><ul><li>p = 1 </li></ul><ul><li>A = 20, 042.4379 </li></ul><ul><li>or </li></ul><ul><li>\$20, 042.44 </li></ul>
17. 18. <ul><li>Danny who offers a Civic that depreciates 0.5% monthly for 3 years. </li></ul><ul><li>A o = 24, 950 </li></ul><ul><li>model = 0.5% -> 0.005 (percent to decimal) </li></ul><ul><li>t = 12 x 3 (because 12 months in one year and it is for 3 years) </li></ul><ul><li>A = ? </li></ul><ul><li>p = 1 </li></ul><ul><li>A = 20, 830.5845 </li></ul><ul><li>or </li></ul><ul><li>\$20, 830.58 </li></ul>
18. 19. <ul><li>Ronnie who offers a Civic that depreciates for 0.15% weekly for 3 </li></ul><ul><li>years. </li></ul><ul><li>A o = 24, 950 </li></ul><ul><li>model = 0.15% -> 0.0015 (percent to decimal) </li></ul><ul><li>t = 52 x 3 (because 52 weeks in one year and it is for 3 years) </li></ul><ul><li>A = ? </li></ul><ul><li>p = 1 </li></ul><ul><li>A = 19, 741.0090 </li></ul><ul><li>or </li></ul><ul><li>\$19, 741.01 </li></ul>
19. 20. <ul><li>We all know that Ronnie is the cheapest one </li></ul><ul><li>So Riolyn will choose him. </li></ul>
20. 21. Problem no. 4 Conics
21. 22. What you see above is a tunnel in Winnipeg that has a length of 162 m and a height of 56 m.
22. 23. <ul><li>If the arch is modeled as an ellipse with the standard form: </li></ul><ul><li>(x - h)² + (y - k)² = 1 </li></ul><ul><li>a² b² </li></ul><ul><li>Determine the equation: </li></ul><ul><li>First, is to use the measures that was given to you. </li></ul><ul><li>Major axis: 2a = 162 </li></ul><ul><li>a= 81 </li></ul><ul><li>Semi-minor axis: b = 54 </li></ul><ul><li>Centre = (0, 0) </li></ul><ul><li>Therefore: (x - h)² + (y - k)² = 1 (x - 0)²+ (y – 0)² = 1 </li></ul><ul><li>a² b² (81)² (54)² </li></ul><ul><li>x² + y² = 1 </li></ul><ul><li>6561 2916 </li></ul>
23. 24. <ul><li>If the arch is modeled as a parabola with the standard form: </li></ul><ul><li>y – k = a(x - h)² </li></ul><ul><li>Determine the equation: </li></ul><ul><li>First state what you know: Vertex = (0 , 54) </li></ul><ul><li>point in the graph = (81, 0) </li></ul><ul><li>Try to plug these values in the equation to find the “a” value. </li></ul><ul><li>y - k = a( x - h )² </li></ul><ul><li>0 - 54 = a(81 - 0) ² </li></ul><ul><li>-54 = a(81) ² </li></ul><ul><li>-54 = 6561 a </li></ul><ul><li>6561 6561 </li></ul><ul><li>a = -54 </li></ul><ul><li>6561 </li></ul><ul><li>y – 54 = -54 x² </li></ul><ul><li>6561 </li></ul>
24. 25. <ul><li>Determine the height of the tunnel at a point 40 m away from the center using </li></ul><ul><li>the equations given above. If the actual height at this point is 40.2 m, </li></ul><ul><li>determine which model makes the closest approximation. </li></ul><ul><li>Ellipse: Parabola: </li></ul><ul><li>x² + y² = 1 y-54 = -54x ² </li></ul><ul><li>6561 2916 6561 </li></ul><ul><li>1600 + y² = 1 y-54 = -54(40) ² </li></ul><ul><li>6561 2916 6561 </li></ul><ul><li>y² = 1 – 0.244 y-54 = -13.169 </li></ul><ul><li>2916 </li></ul><ul><li>y=40.83 </li></ul><ul><li>y² = 0.756 </li></ul><ul><li>2916 </li></ul><ul><li>√ y² = √ 2204.496 </li></ul><ul><li>y =46.952 </li></ul>
25. 26. <ul><li>Ellipse = 46.952 </li></ul><ul><li>Parabola = 40.83 </li></ul><ul><li>Which makes parabola the closest one to the actual height. </li></ul>