A paddle on an antique windmill needs to be visually inspected by a repairmen while the windmill is operating. The paddle can be seen by the repairman when it is at a height of 1.9m on its downward path. If the height of the paddle can be represented by the function, y=1.2764 sin(1.2592x-0.01458)+1.597448, where y is the height of the paddle in meters and x is the time in seconds, the repairman will see the paddle for the third time at _____ seconds?
About 3.41 seconds.
Hmmm. I may be wrong, but I get
1.2764 sin(1.2592x-0.01458)+1.597448 = 1.9
1.2764 sin(1.2592x-0.01458) = 0.31
sin(1.2592x-0.01458) = 0.24
1.26x - 0.015 = 0.25
1.26x = 0.265
x = .196
On the downward track, that means x = pi-1.96 = 2.95
That's the first appearance. Add 4pi to that for the 3rd time.
Looks like it's up to you, sam, to check my work.
To find the time when the repairman sees the paddle for the third time, we need to find the values of x that correspond to y = 1.9m for the third occurrence.
The equation y = 1.2764 sin(1.2592x-0.01458) + 1.597448 represents the height of the paddle at any given time.
To solve for x when y = 1.9m, we can rearrange the equation to:
1.9 = 1.2764 sin(1.2592x - 0.01458) + 1.597448
Subtracting 1.597448 from both sides:
0.302552 = 1.2764 sin(1.2592x - 0.01458)
Dividing both sides by 1.2764:
0.2372 = sin(1.2592x - 0.01458)
To find the angle whose sine equals 0.2372, we can use the inverse sine function (sin^-1) with a calculator. So, sin^-1(0.2372) ≈ 13.835 degrees (rounded to three decimal places).
Now we have:
1.2592x - 0.01458 = 13.835 degrees
To convert the angle to radians, we multiply by π/180:
1.2592x - 0.01458 = 13.835 * π/180
Simplifying:
1.2592x - 0.01458 = 0.241372434
Adding 0.01458 to both sides:
1.2592x = 0.255952434
Dividing both sides by 1.2592:
x ≈ 0.203 seconds (rounded to three decimal places)
Therefore, the repairman will see the paddle for the third time at approximately 0.203 seconds.