This question tests understanding of how to model periodic phenomena using trigonometric functions and identify key parameters: amplitude, midline, period. For a periodic function with maximum value M and minimum value m, the amplitude is A = (M - m)/2 (half the vertical distance from min to max), the midline is D = (M + m)/2 (the average of max and min), and these determine the vertical scaling and shift of the function. Given maximum value M = 30 and minimum value m = 10, the amplitude is A = (30 - 10)/2 = 20/2 = 10, which represents the distance from the midline to either the peak or trough. Choice C is correct because it properly calculates amplitude as (max-min)/2. Choice B uses the full range (max - min) as the amplitude instead of half the range (max - min)/2, confusing peak-to-peak distance with amplitude. Remember that amplitude is half the vertical distance from minimum to maximum, not the full distance—a common error is to use max - min when you should use (max - min)/2. Key to periodic function modeling: always find amplitude as (max - min)/2 and midline as (max + min)/2 first, then determine period from the problem context, calculate B = 2π/P, and choose sine or cosine based on where the function starts at x = 0 or t = 0.

This question tests understanding of how to model periodic phenomena using trigonometric functions and identify key parameters: amplitude, midline, period. The period P of a periodic function is the horizontal distance required to complete one full cycle, and it relates to the coefficient B in y = A·sin(Bx) + D through the formula B = 2π/P, with larger B values producing shorter periods (more frequent oscillations). Since the phenomenon completes 1 cycle in 40 seconds, the period is P = 40. Choice B is correct because it correctly applies B = 2π/P to recognize that the period is the time for one full rotation. Choice A uses incorrect arithmetic, calculating half the time (40/2 = 20) instead of the full cycle time of 40 seconds. For period and B: larger B values mean shorter periods (more frequent oscillations), and the relationship is always B = 2π/P where P is the period, so period and B are inversely related. Key to periodic function modeling: always find amplitude as (max - min)/2 and midline as (max + min)/2 first, then determine period from the problem context, calculate B = 2π/P, and choose sine or cosine based on where the function starts at x = 0 or t = 0.

This question tests understanding of how to model periodic phenomena using trigonometric functions, specifically choosing the appropriate function form based on initial conditions. When modeling with trigonometric functions, choose sine if the phenomenon starts at the midline (moving upward for A > 0), and choose cosine if it starts at a maximum (for A > 0) or minimum (for A < 0), matching the initial condition at t = 0. With amplitude A = 15 (radius), midline D = 18 (center height), period P = 40 giving B = 2π/40 = π/20, and starting at the lowest point (minimum), we use negative cosine to get the function h(t) = -15·cos(πt/20) + 18. Choice C is correct because it properly constructs the function with all parameters and uses negative cosine since the rider starts at the minimum position. Choice A uses positive cosine, which would start at the maximum (top of the wheel), not the minimum as specified in the problem. Choosing between sine and cosine: if the function starts at a minimum, use cosine with negative A; if it starts at a maximum, use cosine with positive A; if it starts at the midline, use sine. To verify your model, check that plugging in t = 0 gives the correct initial value: h(0) = -15·cos(0) + 18 = -15(1) + 18 = 3 meters, which is indeed the lowest point.

This question tests understanding of how to model periodic phenomena using trigonometric functions and identify key parameters: amplitude, midline, period. When modeling with trigonometric functions, choose sine if the phenomenon starts at the midline (moving upward for A > 0), and choose cosine if it starts at a maximum (for A > 0) or minimum (for A < 0), matching the initial condition at t = 0 or x = 0. At t=0, the depth is at midline and increasing, which matches the sine function starting at 0 (midline) with positive slope for A>0. Choice A is correct because it properly identifies that sine starts at midline and increases. Choice B confuses the starting point, stating sine starts at maximum when it actually starts at midline. Choosing between sine and cosine: if the function starts at the midline and increases, use sine with positive A; if it starts at a maximum, use cosine with positive A; if it starts at a minimum, use cosine with negative A or sine with a phase shift. To verify your model, check that plugging in x = 0 gives the correct initial value, and that the function reaches its maximum and minimum at the expected values of midline + amplitude and midline - amplitude.

This question tests understanding of how to model periodic phenomena using trigonometric functions and identify key parameters: amplitude, midline, period. The period P of a periodic function is the horizontal distance required to complete one full cycle, and it relates to the coefficient B in y = A·sin(Bx) + D through the formula B = 2π/P, with larger B values producing shorter periods (more frequent oscillations). Since B = π/6, the period is P = 2π/(π/6) = 12. Choice B is correct because it correctly applies B = 2π/P rearranged to P = 2π/B. Choice A incorrectly calculates P, using 2π/B divided by 2 or similar error instead of the correct P = 2π/B = 12. For period and B: larger B values mean shorter periods (more frequent oscillations), and the relationship is always B = 2π/P where P is the period, so period and B are inversely related. Key to periodic function modeling: always find amplitude as (max - min)/2 and midline as (max + min)/2 first, then determine period from the problem context, calculate B = 2π/P, and choose sine or cosine based on where the function starts at x = 0 or t = 0.

This question tests understanding of how to model periodic phenomena using trigonometric functions and identify key parameters: amplitude, midline, period. The period P of a periodic function is the horizontal distance required to complete one full cycle, and it relates to the coefficient B in y = A·sin(Bx) + D through the formula B = 2π/P, with larger B values producing shorter periods (more frequent oscillations). Since the phenomenon completes 2 cycles in 1 second, the period is P = 1/2, and therefore B = 2π/P = 2π/(1/2) = 4π. Choice C is correct because it correctly applies B = 2π/P. Choice B incorrectly calculates B, using 2π instead of the correct B = 2π/P = 4π. For period and B: larger B values mean shorter periods (more frequent oscillations), and the relationship is always B = 2π/P where P is the period, so period and B are inversely related. Key to periodic function modeling: always find amplitude as (max - min)/2 and midline as (max + min)/2 first, then determine period from the problem context, calculate B = 2π/P, and choose sine or cosine based on where the function starts at x = 0 or t = 0.

This question tests understanding of how to model periodic phenomena using trigonometric functions and identify key parameters: amplitude, midline, period. When modeling with trigonometric functions, choose sine if the phenomenon starts at the midline (moving upward for A > 0), and choose cosine if it starts at a maximum (for A > 0) or minimum (for A < 0), matching the initial condition at t = 0 or x = 0. With amplitude A = 12, midline D = 14, period P = 24 giving B = 2π/24, and starting at maximum, we use cosine to get the function h(t)=12cos((2π/24)t)+14. Choice B is correct because it properly constructs function with all parameters. Choice A chooses sine when cosine is appropriate based on the initial condition—the function starts at a maximum, which corresponds to cosine. Choosing between sine and cosine: if the function starts at the midline and increases, use sine with positive A; if it starts at a maximum, use cosine with positive A; if it starts at a minimum, use cosine with negative A or sine with a phase shift. To verify your model, check that plugging in x = 0 gives the correct initial value, and that the function reaches its maximum and minimum at the expected values of midline + amplitude and midline - amplitude.

This question tests understanding of how to model periodic phenomena using trigonometric functions and identify key parameters: amplitude, midline, period. The general form y = A·cos(B(x - C)) + D has four parameters: A is amplitude, D is midline, B determines period via P = 2π/B, and C is phase shift (horizontal displacement). With amplitude A = 3, midline D = 12, period P = 365 giving B = 2π/365, and starting at maximum at d=172, we use cosine to get the function D(d)=3cos((2π/365)(d-172))+12. Choice A is correct because it properly constructs function with all parameters. Choice B uses the full range (max - min) as the amplitude instead of half the range (max - min)/2, confusing peak-to-peak distance with amplitude. Choosing between sine and cosine: if the function starts at the midline and increases, use sine with positive A; if it starts at a maximum, use cosine with positive A; if it starts at a minimum, use cosine with negative A or sine with a phase shift. To verify your model, check that plugging in x = 0 gives the correct initial value, and that the function reaches its maximum and minimum at the expected values of midline + amplitude and midline - amplitude.

This question tests understanding of how to model periodic phenomena using trigonometric functions and identify key parameters: amplitude, midline, period. The period P of a periodic function is the horizontal distance required to complete one full cycle, and it relates to the coefficient B in y = A·sin(Bx) + D through the formula B = 2π/P, with larger B values producing shorter periods (more frequent oscillations). Since the phenomenon completes 440 cycles in 1 second, the period is P = 1/440, and therefore in 0.01 seconds, the number of cycles is 440 * 0.01 = 4.4. Choice A is correct because it correctly applies the frequency to the time interval. Choice B confuses period with frequency, using 44 when the correct calculation is frequency times time. For period and B: larger B values mean shorter periods (more frequent oscillations), and the relationship is always B = 2π/P where P is the period, so period and B are inversely related. Key to periodic function modeling: always find amplitude as (max - min)/2 and midline as (max + min)/2 first, then determine period from the problem context, calculate B = 2π/P, and choose sine or cosine based on where the function starts at x = 0 or t = 0.

This question tests understanding of how to model periodic phenomena using trigonometric functions and identify key parameters: amplitude, midline, period. For a periodic function with maximum value M and minimum value m, the amplitude is A = (M - m)/2 (half the vertical distance from min to max), the midline is D = (M + m)/2 (the average of max and min), and these determine the vertical scaling and shift of the function. The midline is the horizontal line halfway between the maximum and minimum: D = (8 + 2)/2 = 10/2 = 5, representing the average or equilibrium value of the phenomenon. Choice C is correct because it correctly finds midline as (max+min)/2. Choice D uses the maximum value as the midline, when the midline should be the average of maximum and minimum: (max + min)/2. Key to periodic function modeling: always find amplitude as (max - min)/2 and midline as (max + min)/2 first, then determine period from the problem context, calculate B = 2π/P, and choose sine or cosine based on where the function starts at x = 0 or t = 0. Remember that amplitude is half the vertical distance from minimum to maximum, not the full distance—a common error is to use max - min when you should use (max - min)/2.