Amplitude: 38. a. − 5.47 = 6.975 − 10.85 = 1.29 37. a. 2 2 = 13.43 A = 19.42 A Vertical Shift: 19.42 + 5.47 = 12.445 2 Vertical Shift: 13.43 + 10.85 = 12.14 ω = 2π 2 365 ω = 2π Phase shift (use = 19.42, = 172): y x 365 φ 19.42 = 6.975 sin ⎛ 2π ⋅172 − ⎞ + Phase shift (use = 13.43, = 172): y x 12.445 φ 13.43 = 1.29 sin ⎛ 2π ⋅172 − ⎞ + 12.14 ⎜ 365 ⎟ ⎜ ⎟ 365 ⎝ ⎠ ⎝ ⎠ φ 6.975 = 6.975 sin ⎛ 2π ⋅172 − ⎞ φ = ⎛ ⋅172 − ⎞ 1.29 1.29 sin 2π ⎜ ⎟ 365 ⎜ 365 ⎟ ⎝ ⎠ ⎝ ⎠ φ 1 = sin ⎛ 344π − ⎞ φ 1 = sin ⎛ 344π − ⎞ ⎜ ⎟ 365 ⎜ 365 ⎟ ⎝ ⎠ ⎝ ⎠ π = − φ 344π φ 2 365 π = 344π − φ 2 365 ≈ 1.39 φ ≈ 1.39 − 1.39 ⎞ + 12.445 or sin ⎛ 2π − 1.39 ⎞ + 12.14 . x x Thus, = y 1.29 Thus, = 6.975 sin ⎛ 2π y ⎜ 365 ⎟ ⎜ 365 ⎟ ⎝ ⎠ ⎝ ⎠ ( = ⎡ − + 6.975 sin 2π 80.75)⎤ 12.445 . y x ⎢ 365 ⎥ (91) b. = 1.29 sin ⎛ 2π −1.39 ⎞ + 12.14 y ⎣ ⎦ ⎜ 365 ⎟ (91) ⎝ ⎠ = ⎛ −1.39 ⎞ + b. 6.975 sin 2π y ≈ 12.37 hours 12.445 ⎜ ⎟ 365 ⎝ ⎠ d. The actual hours of sunlight on April 1, ≈ 13.67 hours 2005 was 13.43 hours. This is close to the predicted amount of 13.67 hours. c. y c. y 20 20 10 10 0 140 280 420 x
37. a. Amplitude: A = 19.42 − 5.47 = 6.975 2 38. a. A = 13.43 − 10.85 = 1.29 2 Vertical Shift: 19.42 + 5.47 = 12.445 2 ω = 2π 365 Phase shift (use y = 19.42, x = 172): 19.42 = 6.975 sin ⎛ 2π ⋅172 − φ ⎞ + 12.445 Vertical Shift: 13.43 + 10.85 = 12.14 2 ω = 2π 365 Phase shift (use y = 13.43, x = 172): 13.43 = 1.29 sin ⎛ 2π ⋅172 − φ ⎞ + 12.14 ⎜ 365 ⎟ ⎜ 365 ⎟ ⎝ ⎠ 6.975 = 6.975 sin ⎛ 2π ⋅172 − φ ⎞ ⎝ ⎠ 1.29 = 1.29 sin ⎛ 2π ⋅172 − φ ⎞ ⎜ 365 ⎟ ⎜ 365 ⎟ ⎝ ⎠ 1 = sin ⎛ 344π − φ ⎞ ⎝ ⎠ 1 = sin ⎛ 344π − φ ⎞ ⎜ 365 ⎟ ⎜ 365 ⎟ ⎝ ⎠ π = 344π − φ 2 365 φ ≈ 1.39 ⎝ ⎠ π = 344π − φ 2 365 φ ≈ 1.39 Thus, y = 6.975 sin ⎛ 2π x − 1.39 ⎞ + 12.445 or Thus, y = 1.29 sin ⎛ 2π x − 1.39 ⎞ + 12.14 . ⎜ 365 ⎟ ⎜ 365 ⎟ ⎝ ⎠ ⎝ ⎠ y = 6.975 sin ⎡ 2π ( x − 80.75)⎤ + 12.445 . ⎢ 365 ⎥ b. y = 1.29 sin ⎛ 2π (91) −1.39 ⎞ + 12.14 ⎣ ⎦ ⎜ 365 ⎟ b. y = 6.975 sin ⎛ 2π (91) −1.39 ⎞ + 12.445 ⎝ ⎠ ≈ 12.37 hours ⎜ 365 ⎟ ⎝ ⎠ ≈ 13.67 hours c. y 20 10 0 140 280 420 x d. The actual hours of sunlight on April 1, 2005 was 13.43 hours. This is close to the predicted amount of 13.67 hours. c. y 20 10