Question 6

Question 6 It’s the final showdown!

Question ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

DO NOT MOVE ON UNTIL YOU HAVE ANSWERED THE QUESTION OR YOU NEED HELP!

Things You Should Know ,[object Object],[object Object]

Finding the Equation of the Room ,[object Object],[object Object],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?

Finding the Equation of the Room ,[object Object],[object Object],The center can be found by finding the midpoint between the two vertices.

Finding the Equation of the Room ,[object Object],[object Object],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

Finding the Equation of the Room ,[object Object],[object Object],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

Finding the Equation of the Room ,[object Object],[object Object],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

Finding the Equation of the Room ,[object Object],[object Object],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

Finding the Equation of the Room ,[object Object],[object Object],Now we can connect the dots to create the ellipse. (3, √48) (3, -√48) c= √48 b=4

Finding the Equation of the Room ,[object Object],[object Object],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

Finding the Equation of the Room ,[object Object],[object Object],Our center is (3,0). Our ‘a’ is 8 and our ‘b’ is 4. (3, √48) (3, -√48) c= √48 b=4

How Long Does She Have? ,[object Object],[object Object],(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)

How Long Does She Have? ,[object Object],(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)

How Long Does She Have? ,[object Object],(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)

The Final Clue ,[object Object],[object Object],[object Object],[object Object],[object Object]

The Final Clue ,[object Object]

The Final Clue ,[object Object],n C 2 =66 so we can plug that in. We can also plug in that r=2 (found from n C 2 ).

The Final Clue ,[object Object],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 Final Clue ,[object Object],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.

The Final Clue ,[object Object],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.

The Final Clue ,[object Object],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

The Final Clue ,[object Object],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.

The Final Clue ,[object Object],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’.

The Final Clue ,[object Object],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!!!!

The Final Clue ,[object Object],[object Object]

Hooray! Mary can now defuse the bomb! Let’s hope she remembered to reject the negative value!

What will happen now that Dr. Ping is in the class?

Tune in next year when we take on the challenge of AP CALCULUS!!!

Question 6

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