Question 6

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Question 6 for the season finale of Double Oh 3.14!

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Question 6

  1. 1. Question 6 It’s the final showdown!
  2. 2. Question <ul><li>This is it! It’s the final question! But wait! What’s that gross smell?! OH NO! Dr. Ping has left a bomb in the middle of the room and it’s created a cloud of expanding toxic gas! We gotta solve the question before Mary faints of suffocation! </li></ul><ul><li>a) If the room is an ellipse with the vertices at (3, -8) and (3, 8) and the distance from the center to the foci of the ellipse is √48, draw and graph and write it’s equation. </li></ul><ul><li>b) If the toxic gas cloud starts at the center of the ellipse and expands at a rate of 1m per minute lengthwise (along major axis) and 0.5m per minute width-wise (along minor axis). How long does Mary have to solve the question and get out? </li></ul><ul><li>c) On the paper she found was the clue to defuse the bomb: </li></ul><ul><li>Solve for “n” algebraically: </li></ul><ul><li>n C 2 =66 </li></ul>
  3. 3. DO NOT MOVE ON UNTIL YOU HAVE ANSWERED THE QUESTION OR YOU NEED HELP!
  4. 4. Things You Should Know <ul><li>Oooh. This is a bit of a doozy! This question contains content from Combinatorics and Conics. </li></ul><ul><li>First let’s find the equation of the room. </li></ul>
  5. 5. Finding the Equation of the Room <ul><li>First off, let’s look at what we know: </li></ul><ul><li>a) If the room is an ellipse with the vertices at (3, -8) and (3, 8) and the distance from the center to the foci of the ellipse is √48, draw and graph and write it’s equation. </li></ul>So we know the two vertices and the distance from the center to the foci. Let’s draw that out on a graph shall we?
  6. 6. Finding the Equation of the Room <ul><li>First off, let’s look at what we know: </li></ul><ul><li>a) If the room is an ellipse with the vertices at (3, -8) and (3, 8) and the distance from the center to the foci of the ellipse is √48, draw and graph and write it’s equation. </li></ul>The center can be found by finding the midpoint between the two vertices.
  7. 7. Finding the Equation of the Room <ul><li>First off, let’s look at what we know: </li></ul><ul><li>a) If the room is an ellipse with the vertices at (3, -8) and (3, 8) and the distance from the center to the foci of the ellipse is √48, draw and graph and write it’s equation. </li></ul>Now we can also find the distance of ‘a’. ‘a’ is equal to half the length of the major axis or the distance from the center to one of the vertices. In our example, a=8
  8. 8. Finding the Equation of the Room <ul><li>First off, let’s look at what we know: </li></ul><ul><li>a) If the room is an ellipse with the vertices at (3, -8) and (3, 8) and the distance from the center to the foci of the ellipse is √48, draw and graph and write it’s equation. </li></ul>Now it turns out that in the ellipse, the length ‘a’ (half of the major axis), ‘b’ (half of minor axis) and ‘c” (distance from center to one of the foci) all make a right angled triangle like so: So now let’s mark in the foci so we have ‘a’ & ‘c’. (3, √48) (3, -√48) c= √48
  9. 9. Finding the Equation of the Room <ul><li>First off, let’s look at what we know: </li></ul><ul><li>a) If the room is an ellipse with the vertices at (3, -8) and (3, 8) and the distance from the center to the foci of the ellipse is √48, draw and graph and write it’s equation. </li></ul>From the right triangle that can be formed from the three lengths, ‘b’ can be found through √(a 2 -c 2 )=b. (3, √48) (3, -√48) c= √48 √ (a 2 -c 2 )=b √ (8 2 -(√48) 2 )=b √ (64-48)=b √ (16)=b 4=b
  10. 10. Finding the Equation of the Room <ul><li>First off, let’s look at what we know: </li></ul><ul><li>a) If the room is an ellipse with the vertices at (3, -8) and (3, 8) and the distance from the center to the foci of the ellipse is √48, draw and graph and write it’s equation. </li></ul>Now we can add in length ‘b’ on the graph. Length ‘b’ goes perpendicular to a from the center in both directions. (3, √48) (3, -√48) c= √48 √ (a 2 -c 2 )=b √ (8 2 -(√48) 2 )=b √ (64-48)=b √ (16)=b 4=b b=4
  11. 11. Finding the Equation of the Room <ul><li>First off, let’s look at what we know: </li></ul><ul><li>a) If the room is an ellipse with the vertices at (3, -8) and (3, 8) and the distance from the center to the foci of the ellipse is √48, draw and graph and write it’s equation. </li></ul>Now we can connect the dots to create the ellipse. (3, √48) (3, -√48) c= √48 b=4
  12. 12. Finding the Equation of the Room <ul><li>First off, let’s look at what we know: </li></ul><ul><li>a) If the room is an ellipse with the vertices at (3, -8) and (3, 8) and the distance from the center to the foci of the ellipse is √48, draw and graph and write it’s equation. </li></ul>Now to start writing the equation. First we have to determine whether it is vertical of horizontal. To figure this out, we look at which way the major axis goes. The major axis goes vertically so this is a vertical ellipse. The standard form equation for a vertical ellipse is: (h,k) is the center. a is half the length of the major axis b is half the length of the minor axis (3, √48) (3, -√48) c= √48 b=4
  13. 13. Finding the Equation of the Room <ul><li>First off, let’s look at what we know: </li></ul><ul><li>a) If the room is an ellipse with the vertices at (3, -8) and (3, 8) and the distance from the center to the foci of the ellipse is √48, draw and graph and write it’s equation. </li></ul>Our center is (3,0). Our ‘a’ is 8 and our ‘b’ is 4. (3, √48) (3, -√48) c= √48 b=4
  14. 14. How Long Does She Have? <ul><li>Alright! Now let’s look at how long Mary has to solve the clue and leave. </li></ul><ul><li>b) If the toxic gas cloud starts at the center of the ellipse and expands at a rate of 1m per minute lengthwise (along major axis) and 0.5m per minute width-wise (along minor axis). How long does Mary have to solve the question and get out? </li></ul>(3, √48) (3, -√48) c= √48 b=4 The easiest way to do this would be to make a chart like so: Length of ‘b’ (m). Length of ‘a’ (m). Time (mins)
  15. 15. How Long Does She Have? <ul><li>b) If the toxic gas cloud starts at the center of the ellipse and expands at a rate of 1m per minute lengthwise (along major axis) and 0.5m per minute width-wise (along minor axis). How long does Mary have to solve the question and get out? </li></ul>(3, √48 (3, -√48) c= √48 b=4 Since the cloud has to fill the room, we just look to see how long it takes for the cloud’s ‘a’ and ‘b’ equal the room’s ‘a’ and ‘b’. 4 8 8 3.5 7 7 3 6 6 2.5 5 5 1.5 3 3 2 4 4 1 2 2 0.5 1 1 Length of ‘b’ (m). Length of ‘a’ (m). Time (mins)
  16. 16. How Long Does She Have? <ul><li>b) If the toxic gas cloud starts at the center of the ellipse and expands at a rate of 1m per minute lengthwise (along major axis) and 0.5m per minute width-wise (along minor axis). How long does Mary have to solve the question and get out? </li></ul>(3, √48) (3, -√48) c= √48 b=4 So she has 8 minutes to solve the clue. Ooh, that’s not a lot of time. Let’s hurry and solve the last part. 4 8 8 3.5 7 7 3 6 6 2.5 5 5 1.5 3 3 2 4 4 1 2 2 0.5 1 1 Length of ‘b’ (m). Length of ‘a’ (m). Time (mins)
  17. 17. The Final Clue <ul><li>c) On the paper she found was the clue to defuse the bomb: </li></ul><ul><li>Solve for “n” algebraically: </li></ul><ul><li>n C 2 =66 </li></ul><ul><li>Alright, so now we have to solve for ‘n’. The first thing we have to do to is recall that the Choose formula is: </li></ul><ul><li>We know what ‘r’ is (2) so let’s plug it in and find ‘n’! </li></ul>
  18. 18. The Final Clue <ul><li>n C r = </li></ul>
  19. 19. The Final Clue <ul><li>n C r = </li></ul>n C 2 =66 so we can plug that in. We can also plug in that r=2 (found from n C 2 ).
  20. 20. The Final Clue <ul><li>n C r = </li></ul>n C 2 =66 so we can plug that in. We can also plug in that r=2 (found from n C 2 ). A factorial is defined as n(n-1)(n-2)(n-3)……1. This means that we can expand n! into n(n-1)(n-2)!.
  21. 21. The Final Clue <ul><li>n C r = </li></ul>n C 2 =66 so we can plug that in. We can also plug in that r=2 (found from n C 2 ). A factorial is defined as n(n-1)(n-2)(n-3)……1. This means that we can expand n! into n(n-1)(n-2)!. The (n-2)! in the denominator and the (n-2)! in the numerator reduce to 1.
  22. 22. The Final Clue <ul><li>n C r = </li></ul>n C 2 =66 so we can plug that in. We can also plug in that r=2 (found from n C 2 ). A factorial is defined as n(n-1)(n-2)(n-3)……1. This means that we can expand n! into n(n-1)(n-2)!. The (n-2)! in the denominator and the (n-2)! in the numerator reduce to 1. Multiply both sides by 2.
  23. 23. The Final Clue <ul><li>n C r = </li></ul>n C 2 =66 so we can plug that in. We can also plug in that r=2 (found from n C 2 ). A factorial is defined as n(n-1)(n-2)(n-3)……1. This means that we can expand n! into n(n-1)(n-2)!. The (n-2)! in the denominator and the (n-2)! in the numerator reduce to 1. Multiply both sides by 2. Expand and move all terms to right side so variable is positive
  24. 24. The Final Clue <ul><li>n C r = </li></ul>n C 2 =66 so we can plug that in. We can also plug in that r=2 (found from n C 2 ). A factorial is defined as n(n-1)(n-2)(n-3)……1. This means that we can expand n! into n(n-1)(n-2)!. The (n-2)! in the denominator and the (n-2)! in the numerator reduce to 1. Multiply both sides by 2. Expand and move all terms to right side so variable is positive Factor like a regular quadratic equation.
  25. 25. The Final Clue <ul><li>n C r = </li></ul>n C 2 =66 so we can plug that in. We can also plug in that r=2 (found from n C 2 ). A factorial is defined as n(n-1)(n-2)(n-3)……1. This means that we can expand n! into n(n-1)(n-2)!. The (n-2)! in the denominator and the (n-2)! in the numerator reduce to 1. Multiply both sides by 2. Expand and move all terms to right side so variable is positive Factor like a regular quadratic equation. Solve for ‘n’.
  26. 26. The Final Clue <ul><li>n C r = </li></ul>n C 2 =66 so we can plug that in. We can also plug in that r=2 (found from n C 2 ). A factorial is defined as n(n-1)(n-2)(n-3)……1. This means that we can expand n! into n(n-1)(n-2)!. The (n-2)! in the denominator and the (n-2)! in the numerator reduce to 1. Multiply both sides by 2. Expand and move all terms to right side so variable is positive Factor like a regular quadratic equation. Solve for ‘n’. Now you have found ‘n’. BUT WAIT!!!!
  27. 27. The Final Clue <ul><li>‘ n’ must be a positive whole number. You can’t find the factorial of a negative number or a decimal. You can try it on your calculator. It will say “ERR:Domain”. </li></ul><ul><li>So therefore, the answer to the question is n=12! </li></ul>
  28. 28. Hooray! Mary can now defuse the bomb! Let’s hope she remembered to reject the negative value!
  29. 29. One last question…
  30. 30. What will happen now that Dr. Ping is in the class?
  31. 31. Tune in next year when we take on the challenge of AP CALCULUS!!!

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