Amplitude: 36. a. Amplitude: −10.53 = 1.61 − 9.08 = 3.11 35. a. 2 2 = = 13.75 15.30 A A Vertical Shift: 13.75 + 10.53 = + = 12.14 Vertical Shift: 15.30 9.08 12.19 2 2 ω ω = 2π = 2π 365 365 Phase shift (use = 13.75, = 172): Phase shift (use = 15.30, = 172): y x y x = ⎛ ⋅172 − φ ⎞ + = ⎛ ⋅172 − φ ⎞ + 13.75 1.61sin 2π 15.30 3.11sin 2π 12.19 12.14 ⎜ ⎟ 365 ⎜ 365 ⎟ ⎝ ⎠ ⎝ ⎠ 1.61 = 1.61sin ⎛ 2π ⋅172 − φ φ 3.11 = 3.11sin ⎛ 2π ⋅172 − ⎞ ⎞ ⎜ 365 ⎟ ⎜ ⎟ 365 ⎝ ⎠ ⎝ ⎠ = ⎛ − φ ⎞ 1 sin 344π φ 1 = sin ⎛ 344π − ⎞ ⎜ 365 ⎟ ⎜ 365 ⎟ ⎝ ⎠ ⎝ ⎠ φ π = 344π − π = − φ 2 365 344π 2 365 φ ≈ 1.3900 φ ≈ 1.39 −1.39 ⎞ + 12.14 or ⎛ 2π − 1.39 ⎞ + 12.19 or x x Thus, = y Thus, = 1.61sin ⎛ 2π y ⎜ 365 ⎟ ⎜ ⎟ 365 ⎝ ⎠ ( ⎝ ⎠ = 1.61sin ⎡ 2π − 80.75)⎤ + y x ( = 3.11sin ⎡ 2π − 80.75)⎤ + 12.19 . y x 12.14 . ⎢ ⎥ 365 ⎢ 365 ⎥ ⎣ ⎦ ⎣ ⎦ (91 = ⎡ − + b. 1.61sin 2π 80.75)⎤ y (91) b. = 3.11sin ⎛ 2π −1.3900 ⎞ + 12.19 12.14 y ⎢ 365 ⎥ ⎜ 365 ⎟ ⎣ ⎦ 0 140 280 420 ≈ x 12.42 hours d. The actual hours of sunlight on April 1, c. y 2005 was 12.43 hours. This is very close to the predicted amount of 12.42 hours. 20 ⎝ ⎠ ≈ 12.74 hours c. 10 y 20
35. a. Amplitude: A = 13.75 −10.53 = 1.61 2 36. a. Amplitude: A = 15.30 − 9.08 = 3.11 2 Vertical Shift: 13.75 + 10.53 = 12.14 2 ω = 2π 365 Phase shift (use y = 13.75, x = 172): 13.75 = 1.61sin ⎛ 2π ⋅172 − φ ⎞ + 12.14 Vertical Shift: 15.30 + 9.08 = 12.19 2 ω = 2π 365 Phase shift (use y = 15.30, x = 172): 15.30 = 3.11sin ⎛ 2π ⋅172 − φ ⎞ + 12.19 ⎜ 365 ⎟ ⎜ 365 ⎟ ⎝ ⎠ 1.61 = 1.61sin ⎛ 2π ⋅172 − φ ⎞ ⎝ ⎠ 3.11 = 3.11sin ⎛ 2π ⋅172 − φ ⎞ ⎜ 365 ⎟ ⎜ 365 ⎟ ⎝ ⎠ 1 = sin ⎛ 344π − φ ⎞ ⎝ ⎠ 1 = sin ⎛ 344π − φ ⎞ ⎜ 365 ⎟ ⎜ 365 ⎟ ⎝ ⎠ π = 344π − φ 2 365 φ ≈ 1.3900 ⎝ ⎠ π = 344π − φ 2 365 φ ≈ 1.39 Thus, y = 1.61sin ⎛ 2π x −1.39 ⎞ + 12.14 or Thus, y = ⎛ 2π x − 1.39 ⎞ + 12.19 or ⎜ 365 ⎟ ⎜ 365 ⎟ ⎝ ⎠ y = 1.61sin ⎡ 2π ( x − 80.75)⎤ + 12.14 . ⎝ ⎠ y = 3.11sin ⎡ 2π ( x − 80.75)⎤ + 12.19 . ⎢ 365 ⎥ ⎢ 365 ⎥ ⎣ ⎦ b. y = 1.61sin ⎡ 2π (91 − 80.75)⎤ + 12.14 ⎣ ⎦ b. y = 3.11sin ⎛ 2π (91) −1.3900 ⎞ + 12.19 ⎢ 365 ⎥ ⎜ 365 ⎟ ⎣ ⎦ ≈ 12.42 hours c. y 20 10 0 140 280 420 x d. The actual hours of sunlight on April 1, 2005 was 12.43 hours. This is very close to the predicted amount of 12.42 hours. ⎝ ⎠ ≈ 12.74 hours c. y 20
62514-0132318806_ISM_02-1.docx