1. A pendulum is connected to a rope 3m long, which is connected to a ceiling 4m high. The angle between its widest swing and vertical hanging position is pi/3. If the pendulum swings out to its widest position for 2 seconds, model the horizontal displacement of the pendulum using a sine function, considering vertical to be x=0.
2. The tides at Cape Capstan change the depth of the water in the harbour. On one day in September, the tides have a high point of approximately 13m at 3pm and 1.6m at 9pm. The sailboat has a draft of 2.5m deep. The captain of the sailboat plans to exit the harbourer at 8:30pm.
a. Write a cosine trigonometric equation that represents the situation
described.
b. Plot an accurate graph 1 cycle.
c. Determine whether the sailboat can exit the harbour safely.
3. The average number of customers, c, at a 24-hour sandwich shop per hour is modelled roughly by the equation c(h)=-5cos[pih/12]+12, with h=0 representing midnight.
a. How many hours per day are there 13 customers per hour, to the
nearest hour?
b. What is the maximum number of customers?
c. What time of day is peak(max) business?
4. A 3m ladder is leaning against a vertical wall such that the angle between the ground and the ladder is 3. What is the exact height that the ladder reaches up the wall?
Please show all the work.
4. A 3m ladder is leaning against a vertical wall such that the angle between the ground and the ladder is pi/3. What is the exact height that the ladder reaches up the wall?
1. If the pendulum swings out to its widest position for 2 seconds .....
???? what do you mean ????
2. I guess you mean low tide is 1.6 m at 9 pm which would make sense if high tide was 13 meters at 3 pm and 11.4 meters from low to high in 6 hours would make sense assuming you are in the Bay of Fundy and not here in Gloucester.
mean water level = (1.6 + 13)/2 = 7.3
half range = amplitude = (13-1.6 )/ 2 = 5.7 = amplitude
period = T = 12 hours
so
d = 7.3 + 5.7 cos ( 2 pi t/12 - phi)
when t = 3, d = 13 the max when cos 0 = 1
so
( 2 pi t/12 - phi) = 0 when t = 3
so phi = pi/2
so
d = 7.3 + 5.7 cos ( pi t/6 - pi/2)
so what is d when t = 8.5
d = 7.3 + 5.7 cos (.917 pi) = 7.3 - 5.7 * 0.966 = 7.3 -5.507 = 1.8 meters :(
fast aground :(
Now you try some.
1. To model the horizontal displacement of the pendulum, we can use the equation of a sine function. The general equation for a sine function is given by y = A*sin(B(x - C)) + D, where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.
In this case, since we want to model the horizontal displacement, we will use the x-axis instead of the y-axis. Let's analyze the given information:
- The length of the rope is 3m.
- The height of the ceiling is 4m.
- The angle between the widest swing and the vertical hanging position is π/3.
- The pendulum swings out to its widest position for 2 seconds.
First, let's find the amplitude (A):
The amplitude represents half the distance between the highest and lowest points of the swing. In this case, since the pendulum swings out to its widest position, the amplitude is equal to the length of the rope, which is 3m.
Next, let's find the frequency (B):
The period of a pendulum is given by T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. In this case, L is the length of the rope, which is 3m, and g is approximately 9.8 m/s^2.
Thus, the period T = 2π√(3/9.8) = 2π√(0.306) ≈ 1.76 seconds.
The frequency is the reciprocal of the period, so B = 1/T = 1/1.76 ≈ 0.568 Hz.
The phase shift (C) is the horizontal displacement at the start of the swing. In this case, the pendulum starts at the vertical hanging position, so there is no horizontal displacement. Therefore, C = 0.
Finally, the vertical shift (D) is zero since the zero point is considered to be at the vertical hanging position.
Therefore, the equation for the horizontal displacement of the pendulum is x = 3*sin(0.568t), where t represents time in seconds.
2. a. To write a cosine trigonometric equation that represents the tides at Cape Capstan, we need to find the amplitude, frequency, and phase shift of the cosine function.
- The high point of the tides is approximately 13m at 3pm.
- The low point of the tides is approximately 1.6m at 9pm.
First, let's find the amplitude (A):
The amplitude represents half the distance between the highest and lowest points of the tide. In this case, the amplitude is half the difference between the high point (13m) and the low point (1.6m), which is (13 - 1.6)/2 = 11.7/2 = 5.85m.
Next, let's find the frequency (B):
Since the tide cycle repeats every 12 hours (from high to high or low to low), the frequency is 1 cycle per 12 hours, or 1/12 cycles per hour.
The phase shift (C) is determined by the time when the tide reaches its highest point. In this case, the high point is at 3pm, so there is a phase shift of 3 hours.
Finally, the vertical shift (D) is zero since the zero point is considered to be at the average water level.
Therefore, the cosine trigonometric equation that represents the tides at Cape Capstan is y = 5.85*cos((1/12)(x - 3)).
b. To plot an accurate graph for one cycle, we can take the x-axis to represent time in hours (with 3pm as the starting point) and the y-axis to represent the water depth in meters.
c. To determine whether the sailboat can exit the harbor safely, we need to consider the draft of the sailboat, which is 2.5m. The sailboat can safely exit the harbor when the water depth is greater than the draft.
Using the equation y = 5.85*cos((1/12)(x - 3)), substitute 8.5 (8:30 pm) for x to find the water depth at that time. If the water depth is greater than 2.5m, then the sailboat can safely exit the harbor.