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Trigonometric Applications: The Tarzan Problem

Trigonometric Applications: The Tarzan Problem. Ms. Alison Heltemes.

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Trigonometric Applications: The Tarzan Problem

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  1. Trigonometric Applications:The Tarzan Problem Ms. Alison Heltemes

  2. Tarzan problem:Tarzan is swinging back and forth on his grapevine. As he swings, he goes back and forth across the river bank, going alternatively over land and water. He decides to model his motion mathematically and starts his stopwatch.Let x be the number of seconds the stopwatch reads and y the number of meters Tarzan is from the river bank. y is positive when he is over water and negative when he is over land.

  3. 1. Sketch a graph of Tarzan’s motion. • Tarzan finds that when x = 2, he is at one end of his swing where y = -23. (Min)When x = 5 he reaches the other end of his swing where y = 17. (Max)

  4. 1. Sketch a graph of Tarzan’s motion. Where would be the point halfway in between these two points? (3.5, -3) So the middle is at -3 and every 1.5 seconds there is a point on the graph. Fill in the points now. And your curve. (-1,17) (.5, -3) (6.5, -3) (8, -23) (9.5, -3) (11,17) (12.5, -3)

  5. 2. Find an equation for Tarzan’s motion. • Middle (D): -3 • Amplitude (A): 20 • Period: 6 seconds • B : π/3 Use the point (2,-23) as your starting point. Find the equation. y= -3-20cos((π/3(x-2)

  6. 3. Predict y when x = 6.3 Substitute 6.3 in for x. y = -3 - 20cos((π/3)(6.3-2)) y = 1.16 meters over the water

  7. Where was Tarzan when he started the stop watch? Substitute 0 in for x. y = -3 - 20cos((π/3)(0-2)) y = 7 meters over the water

  8. Find the first three times when Tarzan was directly over the river bank. Substitute 0 in for y. 0 = -3 - 20cos((π/3)(x-2)) Step 1: Get trig function by itself 3 = - 20cos((π/3)(x-2)) -3/20 = cos((π/3)(x-2)) Step 2: Determine which quadrants Cosine is negative in the 2nd and 3rd quadrants

  9. Find the first three times when Tarzan was directly over the river bank. -3/20 = cos((π/3)(x-2)) Step 3: Find 1st angle using inverse Cos-1(-3/20) = 1.72 Step 4: Find 2nd angle using 1st as an aid 2nd angle = -1.72

  10. Find the first three times when Tarzan was directly over the river bank. -3/20 = cos((π/3)(x-2)) Step 5: Set ( ) = each solution + 2πn 1. π/3(x-2) = 1.72 + 2πn 2. π/3(x-2) = -1.72 + 2πn Step 6: Solve for x 1. π/3(x-2) = 1.72 + 2πn 2. π/3(x-2) = -1.72 + 2πn Divide by π/3 or multiply by the reciprocal 3/π 1. x-2 = 1.64 + 6n 2. x-2 = -1.64 + 6n Add 2 to the like term 1. x= 3.64 + 6n 2. x= .36 + 6n

  11. Find the first three times when Tarzan was directly over the river bank. Step 7: Check to see if solution is within 1 positive period length One period length is from 0 to 6. 1. x= 3.64 + 6n yes 2. x= .36 + 6n yes Step 8: Answer Question Tarzan was over the riverbank at .36 seconds, 3.64 seconds and (.36 + 1period length which is 6) = 6.36 seconds

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