I have finished the questions for the following, but could someone please check to see if my answers are right because there are no answers for these questions since it's a sheet that my teacher gave us.
Ok, just let me know where my errors are and what is the correct answer if applicable.
Thank you in advance!
1. Write a sinusoidal function for the function with a period of 6. The function has a max of 3 at x=2 and a low point of -1.
My answer: y=2sin[(pi/3)(x)] +1
2. Write a sinusoidal function for the function with a period of 5. The function has a max of 7 at x=1.
My answer: y=3.5sin[(2pi/5)(x+0.25)]
3. When you board a Ferris wheel your feet are 1 foot off the ground. At the highest point of the ride, your feet are 99 feet above the ground. It takes 30 seconds for the ride to complete one revolution. Write a sinusoidal function for your height above the ground at t seconds after the ride starts.
My answer: y=49sin[(pi/15)(x-30)+50]
4. At high tide the water level at a particular boat dock is 9 feet deep. At low tide, the water is 3 feet deep. On a certain day the low tide occurs at 3 am and the high tide occurs at 9 am. Find an equation for the height of the tide at time t.
My answer: y=3cos(pi/12x)+6
5. Jessie has a pulse rate of 73 beats per minute and a blood pressure of 121 over 85. If Jessie's blood pressure can be modeled by a sinusoidal function, find an equation of this sinusoid.
My answer: y=0.7sin(146pix)
6. As the paddlewheel turned, a point on the paddle blade moved back in such a way that its distance, d, from the water's surface was a sinusoidal function of time. When a stopwatch read 4 seconds the point was at it's highest, 16ft about the water's surface. The wheel's diameter was 18 ft. and it completed a revolution every 10 second.
My answer: y=9sin[pi/5(x-1.5)]+7
7. The number of sunspots counted in a given year varies periodically from a minimum of about 1 per year to about 110 per year. Between the maximums that occurred in the years 1750 and 1948, there were 18 complete cycles.
my answer: y=54.5sin(2pi/11x)+55.5
8. You seek treasure that is buried in the side of a mountain. The mountain range is a sinusoidal cross section. The valley to the left is filled with water to a depth of 50 meter, and the top of the range is 150 meters above water level. You set up an x-axis at water level and a y axis 200 meters to the right of the deepest part of the water. The top of the mountain is at x=400 meter.
my answer: y=225sin[pi/100(x-200)]+175
9. A creatures body temperature is varying sinusoidally with time. 35 minutes after they start timing it reaches a high of 120 degrees F and a 20 minutes after that its next low is 104 degrees F.
my answer: y=8sin[pi/20(x-25)]+112
10. A ferris wheel is 50 ft in diameter, with the center 60 ft above the ground. You enter from a platform at the 3oclock position. It takes 80 sec to complete one revolution. Find the equation that gives you your height when you entered the ferris wheel above the ground at t time. (t=0 when you entered).
my answer: y=-25sin(pi/40x)+35
Okay, thanks again for whoever who helps! I really really appreciate it!
#1
you will need a phase shift, the way it stands your max would occur at x=1.5, but you want it to happen at x=2
the entire graph would have to be shifted to the right by 1/2 units, so ...
y = 2sin (pi/3)(x - 1.5) + 1
#2
You should sub in the given values in your answer equation to see if you get the correct result.
e.g. when x=1 in yours you would get
y = 3.5sin[(2pi/5)(1.25)]
= 3.5 not the 7 we need
so make the altitude 7
y = 7sin[(2pi/5)(x+0.25)]
#3
check the brackets, also since it takes 30 sec to complete a rotation, a phase shift of 30 sec would accomplish nothing.
you had : y=49sin[(pi/15)(x-30)+50]
I would say y=49sin(pi/15)(x-7.5) + 50
check when x=0, y = 1
when x= 15, we should be at the top
y = 49sin(pi/15)(15-7.5) + 50 = 99
do the same kind of checking for you others
sub in the given data to see if it works, then adjust accordingly
You seem to make errors in the phase shift
#4
from low tide to high tide is 6 hours, but that is only half a cycle, wouldn't it have to back to low tide to have a full period ? so k= pi/6, not pi/12
e.g. if x= 6 (9:00 am), your result would give us
y = 3cos(pi/12)(6)+6 = 6, not the 9 we need
so y = -3cos(pi/6)x + 6
test:
x=0, y = 3
x=6 t = 9
To check your answers, let's go through each question one by one.
1. Write a sinusoidal function for the function with a period of 6. The function has a max of 3 at x=2 and a low point of -1.
Your answer: y=2sin[(pi/3)(x)] +1
To check this answer, we need to determine if the period, maximum, and minimum are correct. The period of a sinusoidal function is given by 2π divided by the coefficient of x inside the sine function. In your answer, the coefficient of x is (π/3), so the period is 2π/((π/3)) = 6. This is correct. The maximum is 3, and the function has shifted up by 1 unit, which is also correct. Therefore, your answer is correct.
2. Write a sinusoidal function for the function with a period of 5. The function has a max of 7 at x=1.
Your answer: y=3.5sin[(2pi/5)(x+0.25)]
To check this answer, we can apply the same method. The period is 5, and the coefficient of x is (2π/5), so the period is 2π/((2π/5)) = 5. This is correct. The maximum is 7, and the function has shifted horizontally by 0.25 units, which is correct. Therefore, your answer is correct.
3. When you board a Ferris wheel your feet are 1 foot off the ground. At the highest point of the ride, your feet are 99 feet above the ground. It takes 30 seconds for the ride to complete one revolution. Write a sinusoidal function for your height above the ground at t seconds after the ride starts.
Your answer: y=49sin[(pi/15)(x-30)+50]
To check this answer, we need to consider the period, minimum, and maximum. The period is 30 seconds, which aligns with the given information. The function has a minimum of 1 and a maximum of 99. However, the function should be centered at the average of the minimum and maximum, which is (99+1)/2 = 50. So the function should be shifted vertically by 50 units, not by 49 units. Therefore, your answer should be y=50+49sin[(pi/15)(x-30)].
4. At high tide, the water level at a particular boat dock is 9 feet deep. At low tide, the water is 3 feet deep. On a certain day, the low tide occurs at 3 am, and the high tide occurs at 9 am. Find an equation for the height of the tide at time t.
Your answer: y=3cos(pi/12x)+6
To check this answer, we can determine if the maximum and minimum align with the given information. The maximum should be 9, and the minimum should be 3. The function is centered around the average of the maximum and minimum, which is (9+3)/2 = 6. So the function should be shifted vertically by 6 units, not by 0 units. Therefore, your answer should be y=6+3cos(pi/12x).
5. Jessie has a pulse rate of 73 beats per minute and a blood pressure of 121 over 85. If Jessie's blood pressure can be modeled by a sinusoidal function, find an equation of this sinusoid.
Your answer: y=0.7sin(146pix)
To check this answer, we need to consider if the amplitude, period, and vertical shift are correct. The amplitude represents half the range of the function, so the amplitude should be (121-85)/2 = 18. The coefficient of x inside the sine function determines the period, which in this case is (146π). Therefore, the period is 2π/(146π) = 1/73, which aligns with the pulse rate of 73 beats per minute. However, the function should be centered vertically at the average of the maximum and minimum, which is (121+85)/2 = 103. The function should be shifted vertically by 103 units, not by 0 units. Therefore, your answer should be y=103+0.7sin(146pix).
I will provide feedback on the remaining answers in another response.