Question

URGENT!!! A farmer has a small windmill on her farm that is 4 m in diameter and its centre is 17 m above the ground. Before it starts spinning, a lady bug lands on the very bottom edge of the propeller that is pointing downward. Once it starts spinning, it takes only 6 seconds for the ladybug to complete two complete cycles around the windmill.

a) What are the amplitude and axis of the curve going to be for the function that
represents this situation? How do you know?
Hint: How are these two values related to this real-world example?
b) Determine the maximum and minimum height from the ground that the ladybug
will have on the windmill. Explain your reasoning.
c) What is the period of the windmill? Explain your reasoning.
D) Determine the equation of the trig function that models the height from the
ground of the ladybug as a function of time.
Be sure to do the following:
• define your variables
• show how you got each of the parameters (a, k, c)
• show how you decided whether to use sin or cos
E) After the two cycles, the ladybug keeps gripping onto the propellor and continues
going around. What is its height two (2) seconds after the second cycle is complete.
Round your answer to the nearest tenth of a metre.

Answer 1

my answer:

a) The amplitude and axis of the curve can be determined from the given information. In this real-world example, the windmill's propeller rotates in a circular motion. The amplitude of the curve represents the distance from the center of the windmill to the highest or lowest point on the propeller. Since the windmill has a diameter of 4 m, the radius (and therefore the amplitude) is half of that, which is 2 m.

The axis of the curve represents the vertical line passing through the center of the windmill. In this case, the windmill's center is 17 m above the ground. Thus, the axis of the curve is a horizontal line at a height of 17 m.

b) To determine the maximum and minimum height of the ladybug on the windmill, we need to consider the amplitude. Since the amplitude is 2 m, the maximum height the ladybug will reach is 17 m + 2 m = 19 m above the ground. The minimum height will be 17 m - 2 m = 15 m above the ground.

c) The period of the windmill represents the time it takes for the ladybug to complete one full cycle around the windmill. From the given information, we know that the ladybug takes 6 seconds to complete two complete cycles. Therefore, the period of the windmill can be calculated by dividing the total time by the number of cycles, which is 6 seconds / 2 = 3 seconds per cycle.

In summary:

a) The amplitude is 2 m, and the axis is a horizontal line at a height of 17 m.

b) The maximum height of the ladybug is 19 m above the ground, and the minimum height is 15 m above the ground.

c) The period of the windmill is 3 seconds per cycle.

there ya go i hope it helped
(real)

Answer 2

A) y = 2 sin(kx) + 17

B) 15 meters

C) 3 seconds

D) y = 2 sin(2π/3 x) + 17

E) y ≈ 18.9

Explanation:

a) The motion of the ladybug on the windmill can be modeled using a sinusoidal function of the form y = a sin(kx) + c, where y represents the height of the ladybug above the ground, x represents time, a represents the amplitude, k represents the frequency, and c represents the vertical shift or axis of the curve.

Since the ladybug is moving up and down in a periodic manner, we can infer that the equation will be a sinusoidal function. The amplitude of the function will be half of the vertical distance that the ladybug travels, which is equal to the radius of the windmill, or 2 meters. The axis of the curve will be the average height of the ladybug, which is equal to the height of the windmill's center, or 17 meters.

Therefore, the equation of the curve is y = 2 sin(kx) + 17.

b) The maximum and minimum height of the ladybug can be found by adding and subtracting the amplitude from the axis of the curve, respectively. Therefore, the maximum height of the ladybug is:

17 + 2 = 19 meters

and the minimum height of the ladybug is:

17 - 2 = 15 meters.

c) The period of the function is the time it takes to complete one full cycle of the sinusoidal motion. In this case, the ladybug completes two cycles in 6 seconds, so the period is:

6 seconds / 2 = 3 seconds.

d) In the equation y = 2 sin(kx) + 17, y represents the height of the ladybug above the ground, x represents time in seconds, a represents the amplitude, k represents the frequency, and c represents the vertical shift of axis of the curve.

To find k, we use the formula:

k = 2π / T,

where T is the period of the function. Substituting T = 3 seconds, we get:

k = 2π / 3.

Therefore, the equation of the curve is:

y = 2 sin(2π/3 x) + 17.

e) After completing two cycles, the ladybug has returned to its initial position, so its height is equal to the axis of the curve, which is 17 meters. Two seconds after completing the second cycle, the ladybug has moved a quarter of a cycle, or 0.75 seconds. Therefore, we can find the height of the ladybug at this time by substituting x = 0.75 into the equation:

y = 2 sin(2π/3 x) + 17

y = 2 sin(2π/3 × 0.75) + 17

y ≈ 18.9