SOLUTION: The equation h(t)=40sin(5π/4 t- π/2)+55 describes the height, H in cm, that the centre of a yo-yo is above the ground, during standard motion, as a time in seconds. Use

Algebra ->  Trigonometry-basics -> SOLUTION: The equation h(t)=40sin(5π/4 t- π/2)+55 describes the height, H in cm, that the centre of a yo-yo is above the ground, during standard motion, as a time in seconds. Use      Log On


   

Question 725616: The equation h(t)=40sin(5π/4 t- π/2)+55
describes the height, H in cm, that the centre of a yo-yo is above the ground, during standard motion, as a time in seconds.
Use the equation to determine the:
(i) length of the string.
(ii) smallest distance between the yo-yo and the ground
(iii) the time taken to complete one swing
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
(i) The length of the string would be the distance between the highest point (the hand) and the lowest point. The distance between the maximum and the minimum of a sin graph is twice the amplitude. The 40 in the equation is the amplitude so the length of the string is 80 cm.

(ii) The minimum value for a sin function (with a positive coefficient like 40) will occur when sin has its minimum value. The minimum value of a sin is -1. Replacing the sin with -1 we can find the minimum height:
40(-1) + 55 = 15 cm

(iii) The time it takes to for one complete swing/cycle is the period of the function. In a sin equation like this one, the period is related to the coefficient of x (called "B") by the equation:
B = 2pi/period
The "B" in our equation is 5pi%2F4 so:
5pi%2F4+=+2pi/period
Now we can solve for the period. Multiplying both sides by the period:
(period)*%285pi%2F4%29+=+2pi
Dividing both sides by 5pi%2F4:
period = 2pi%2F%285pi%2F4%29
which simplifies to:
period = 8/5
So it takes 8/5 (or 1.6) seconds for one complete cycle.