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Question 2 for Episode 2 of Double Oh 3.14!

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- 1. Question 2 Where should Mary hit the ball?
- 2. Question <ul><li>Agent Double OH 3.14 has engaged herself in a ping pong match with the Ping Pong Fiend. Mary isn’t able to hit the ball back but she notices something. The ball makes the same curve each time it goes over the net and it makes a sinusoidal curve. Mary notices that: </li></ul><ul><li>The ball reaches a maximum height of 48 cm right above the net. </li></ul><ul><li>The ball hits the table 80cm away from the net before it goes over. </li></ul><ul><li>Sketch a sinusoidal graph to show the path of the ping pong ball going towards Mary. </li></ul><ul><li>Write two equations to represent this function (one in sine and one in cosine). </li></ul><ul><li>Determine how high Mary’s paddle has to be in order to hit the ping pong ball if she extends her paddle 106cm from the net? </li></ul><ul><li>Oh no! Mary has become paralyzed by the Ping Pong Fiend. Her paddle can only reach a height of 15cm. How far must she reach to hit the winning shot? </li></ul>
- 3. Question <ul><li>But wait! Seeing how far she has to hit isn’t all! She has to be able to hit the ball. </li></ul><ul><li>There are three types of shots: Slice, Lob and a Smash. </li></ul><ul><li>Mary has a 1/15 shot of hitting a smash, a 6/15 chance of hitting a slice and a 7/15 chance of hitting a lob. </li></ul><ul><li>If Mary hits a lob, the Fiend has a 7/11 shot of returning it. </li></ul><ul><li>If Mary hits a slice, the Fiend has a 3/11 shot of returning it. </li></ul><ul><li>If Mary hits a smash, the Fiend has a 10/11 shot of not returning it. </li></ul><ul><li>What is the probability that Mary smashed the ball given the fact that the Fiend did not return the ball? </li></ul>
- 4. DO NOT MOVE ON UNTIL YOU HAVE ANSWERED THE QUESTION OR YOU NEED HELP!
- 5. Things You Should Know <ul><li>Alright so obviously this question will deal with Trigonometry and Probability. </li></ul><ul><li>We’ll start with the Trigonometric part. </li></ul>
- 6. Trigonometric Part <ul><li>The trigonometric part of this question is obviously found in the sinusoidal graph and equations. The first part of this question is asking for the graph. </li></ul><ul><li>Sketch a sinusoidal graph to show the path of the ping pong ball going towards Mary. </li></ul>
- 7. Graphing (Trigonometry) Ok, let’s start graphing this. The first thing you would have to choose is where the x-axis and the y-axis will go. In this graph, we’ll let the surface of the table be the x-axis (0 cm from the ground.) We’ll let the y-axis be the position of the net (0 cm from the net).
- 8. Graphing (Trigonometry) Next, we can plot the maximum and minimum points. The maximum y-value of this graph-to-be is 48 cm (info from question) and this happens when it is directly over the net. In our graph, the y-axis represents the position of the net. The minimum value of this graph is 0 cm (hits the table.). This happens 80cm away from the net in both directions.
- 9. Graphing (Trigonometry) Next we find the sinusoidal axis. That is just a fancy set of words that simply mean the average. The sinusoidal axis is not in a solid line because it is not actually part of the graph. To find it, the sinusoidal axis is the same as the midpoint line between the maximum and the minimum.
- 10. Graphing (Trigonometry) Now we can graph the points that are on the sinusoidal axis. These points are located halfway horizontally between the maximum points and the minimum points. In this case, the half distance between the maximum point and the minimum points is 40. This means that you would create points on the graph that are on the sinusoidal axis that are 40 units away from the x values of each maximum and minimum point.
- 11. Graphing (Trigonometry) Now we connect all the points we have created with a smooth curve. You graph may not look like this one but it should look close to this.
- 12. Trigonometric Part <ul><li>Now that we have the graph, we can do part b) of the question. </li></ul><ul><li>Write two equations to represent this function (one in sine and one in cosine). </li></ul>
- 13. Things You Should Know <ul><li>Before we start making the equations, we should start by explaining each equation. </li></ul>
- 14. Things You Should Know <ul><li>The basic form for the sine function is: </li></ul><ul><li>f(x)=Asin(B(x-C))+D </li></ul><ul><li>The basic form for the cosine function is: </li></ul><ul><li>f(x)=Acos(B(x-C))+D </li></ul><ul><li>In both these forms, the parameters A, B, C and D mean the same thing. </li></ul>
- 15. Things You Should Know <ul><li>Parameter A is the vertical stretch/compression. This value is also equal to the amplitude of the wave (length from sinusoidal axis to a max/min point). </li></ul><ul><li>Parameter B is equal to the horizontal stretch/compression. This value is equal to 2 π /(period of wave). </li></ul><ul><li>Parameter C is equal to the horizontal shift of the wave. </li></ul><ul><li>Parameter D is equal to the vertical shift of the wave. This value also equal to the value of the sinusoidal axis. </li></ul><ul><li>The “easiest” order to find the parameters is D, A, B, C. You don’t have to follow this order. This is based on opinion. </li></ul>f(x)=Asin(B(x-C))+D f(x)=Acos(B(x-C))+D
- 16. Things You Should Know <ul><li>If you are writing both equations for the same wave, the values of parameters A, B and D will be the same. The only value that will be different between the two equations is parameter C. </li></ul><ul><li>This is because when θ =0, the value of sine is 0 and the value of cosine is 1. This means that the graphs of sine and cosine start at different points. </li></ul>f(x)=Asin(B(x-C))+D f(x)=Acos(B(x-C))+D
- 17. Writing the Equation (Trigonometry) So let’s start with writing the sine equation. First we’ll find parameter D. Why not “A” you ask? D is just easier to find that’s all. So D is equal to the value of the sinusoidal axis. The equation of the sinusoidal axis in this case is y=24. This means the value of parameter D is 24.
- 18. Writing the Equation (Trigonometry) Next we’ll find parameter A. This value is the same as the amplitude. The amplitude is the distance between a max/min point and the sinusoidal axis. In this case, the distance between the sinusoidal axis and any one max/min point is 24. This means that the value of parameter A is 24. There is something else to know about A but we will talk about that after we find C.
- 19. Writing the Equation (Trigonometry) Next we’ll find parameter B. Parameter B is the same as the 2 π /(period of wave). The period of the wave is the length it takes for the wave to repeat. To find it, pick two points that are on the same part of the curve. This could be two consecutive mins, two consecutive maxes, or any two consecutive points that are on the same part of the graph. (ex. Any pair of same coloured points: black, purple, orange, red)
- 20. Writing the Equation (Trigonometry) To find parameter B, we’ll use the red dots (the min. values). The distance between the dots is 160. This means that the period of the wave is 160. This, however, does not mean that parameter B is 160. Parameter B is 2 π /(period of wave). Parameter B is equal to 2 π /160. Simplifying this fraction is optional. Keeping it has it’s benefits because from that, you can immediately see the period of the wave.
- 21. Writing the Equation (Trigonometry) Now for parameter C. Because the value of C is different for the sine equation and the cosine equation, we need to find them separately. First we’ll start with sine.
- 22. Writing the Equation (Trigonometry) We know that the graph of y=sin(x) starts at a “mid-point” in relation to the y-axis (It doesn’t start at the max value or the min value.) To find parameter C, we just look at how far any “mid-point” is from the y-axis. The value of C is different depending on what point you choose. In this example, we’ll use the first positive “mid-point”. This point is 40 units to the right of the y-axis so parameter C for the sine equation is 40.
- 23. Writing the Equation (Trigonometry) Now for cosine. We know that the graph of y=cos(x) starts at a maximum value. To find parameter C you look at how far any max value is from the y-axis. In this case we will use the closest max value to the y-axis. This value also happens to be on the y-axis so the value of parameter C is 0. You could also pick a minimum point. We will discuss that shortly.
- 24. The Issue with A (Sine) This isn’t really an issue but it could mess up your equation. When you choose your point that you are looking at, look at the way the graph passes through that point. The regular sine curve goes from the lower left to the upper right passing through this point. However, if the curve goes from the upper left to the lower right through your point, that means this graph has been flipped vertically and the amplitude of the wave is negative.
- 25. The Issue with A (Cosine) The issue with A being negative also appears in the cosine equation. The regular equation starts with a maximum point so if you pick a maximum point, your amplitude will be positive. However, you could also pick a minimum point to find the value of parameter C. However, picking a minimum point means that the graph of y=cos(x) has been flipped vertically. When you pick a minimum point to use as a reference, your amplitude will be negative.
- 26. Building the Equations Cosine A= 24 B= 2 π /160 C= 0 D= 24 Now we can start building the equations. Just take the basic form and input in all the parameters we just found. Sine A= -24 (explanation why A is negative: slide 24) B= 2 π /160 C= 40 D= 24 Your equations may be different depending on your parameters. However, the only parameter that should be different is parameter C.
- 27. Trigonometric Part <ul><li>Time to move on to part c). </li></ul><ul><li>Determine how high Mary’s paddle has to be in order to hit the ping pong ball if she extends her paddle 106cm from the net? </li></ul>
- 28. Solving the Equation (Trigonometry) This one is easy once you have the equation. Just plug and play! We’ll solve this equation using both the sine equation and the cosine equation we just built. She wants to hit the ball 106 cm from the net. We need to find the height at which the ball is when it is that far from the net. We let x=106 and we solve for y. Cosine Sine Mary has to have her paddle 11.46 cm from the surface of the table to hit the ball.
- 29. Trigonometric Part <ul><li>Time to move on to part d). </li></ul><ul><li>Oh no! Mary has become paralyzed by the Ping Pong Fiend. Her paddle can only reach a height of 15cm. How far must she reach to hit the winning shot? Answer in interval of [0, 2 π ]. </li></ul>
- 30. Solving the Equation (Trigonometry) This one is like part c). However, now we are looking for the distance from the net, meaning we let f(x)=15 and we are looking for x. As you can see in the equation, the part in brackets looks pretty intimidating. For now, we can let θ =whatever is in the brackets and solve for θ . Cosine Sine
- 31. Solving the Equation (Trigonometry) Cosine Sine We are only using a decimal approximation but on your calculator you get a lot more decimal places. To keep this exact value, press [STO->][ALPHA][MATH] on your calculator to store that value as “A”. Use this exact value in your calculation for “x”. To use this value, press [ALPHA][MATH] on your calculator.
- 32. Solving the Equation (Trigonometry) Congratulations! You have found θ ! However, there are actually two answers for θ in each case. When sine is positive, the angle could be in quadrants 1 and 2. When cosine is negative, the angle could be in quadrants 2 and 3. To complete the answer, we need to find the related angles as well. The sine related angles are in green and the cosine related angles are in red. To find the other angle from the sine equation just take π - θ . To find the other angle from the cosine equation, take 2 π - θ . Great! We found all the θ values. Now we have to find “x”! Remember? We substituted θ for the part with the “x” earlier.
- 33. Solving the Equation (Trigonometry) Sine
- 34. Solving the Equation (Trigonometry) Cosine So now we have two values of “x”. x= 49.7886, 110.2114
- 35. So how do we know which one to eliminate (if there is one to eliminate)?
- 36. Solving the Equation (Trigonometry) There is an answer we need to eliminate. Remember that this is a game of ping pong. You must let the ball hit the table first before you return. If we look at the graph, the ball hits the table at (80, 0). This means that the x value we are looking for is larger than 80. We can eliminate 49.7886 meaning that Mary must reach her arm out 110.2114 cm from the net to hit the winning shot!
- 37. Probability Part <ul><li>Time to move on to the last part. </li></ul><ul><li>There are three types of shots: Slice, Lob and a Smash. </li></ul><ul><li>Mary has a 1/15 shot of hitting a smash, a 6/15 chance of hitting a slice and a 7/15 chance of hitting a lob. </li></ul><ul><li>If Mary hits a lob, the Fiend has a 7/11 shot of returning it. </li></ul><ul><li>If Mary hits a slice, the Fiend has a 3/11 shot of returning it. </li></ul><ul><li>If Mary hits a smash, the Fiend has a 10/11 shot of not returning it. </li></ul><ul><li>What is the probability that Mary smashed the ball given the fact that the Fiend did not return the ball? </li></ul>
- 38. What are the Chances? There are two ways to do this question. One with a drawing and one using equations. It doesn’t matter which way you look at it. It’s like Mr.K’s block of wood. They are just different perspectives on the same question. First we’ll look at the method with drawing a tree. Well we know that there are three different types of shots Mary can make so let’s list them on the tree. Mary could also miss.
- 39. What are the Chances? We also know that after Mary makes her shot the Fiend will either return it or not return it. So let’s mark that down too!
- 40. What are the Chances? We also know some probabilities. Let’s mark it! P(Smash) = 1/15 P(Not returned if smashed) = 10/11 P(Lob) = 7/15 P(Returned if lobbed) = 7/11 P(Slice) = 6/15 P(Returned if sliced) = 3/11
- 41. What are the Chances? We can fill in the rest of the probabilities like so: P(Miss) = 1/15 (remaining part out of 15 from the 4 possibilities Mary has) P(returns if smashed)= 1/11 (because 10/11 chance of a miss) P(not returned if sliced) = 8/11 (because 3/11 chance of a hit) P(not returned if lobbed) = 4/11 (because 4/11 chance of a hit) P(returned if miss) = 0/11 (can’t return a ball that isn’t going towards you) P(not returned if miss) = 11/11 (can’t miss a ball that wasn’t returned to you)
- 42. What are the Chances? Now that we can calculate the probabilities of each scenario happening by multiplying the probabilities as we travel down each branch.
- 43. What are the Chances? To calculate the probability that Mary smashed the ball given the fact that the Fiend missed, the formula is: All we do is plug in the values and we find our probability!
- 44. What are the Chances? or P(smash|not returned)≈10.3093%
- 45. So the chances of Mary hitting the ball is approximately 10.3093%!
- 46. Click the word “Enjoy~!” At the beginning to move on to Episode 3!

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