Can you help me solve this problem? I really don't know what to do. I solve problems like for example the rate of change of the elevation but I just encountered this kind of problem so I really don't know what to do. Please help me
The endpoints of a movable rod of length 1 meter have coordinates (x,0) and (0,y). The position of the end on the x-axis is x(t)= (3/5) sin (pi*t) where t is time in seconds.
a) Find the time of 1 complete cycle of the rod.
b) What is the lowest point reached by the end of the rod on the y-axis?
c) Find the speed of the y-axis endpoint when the x-axis endpoint is (3/10, 0)
(a) the period of sin(kt) is 2π/k, so we have period 2π/π = 2 seconds.
(b) since the minimum of sin(kt) = -1, our minimum is -3/5
(c) since the length of the rod is 1,
x^2+y^2 = 1
That means that
y^2 = 1/25 (16 + 9cos^2(πt))
when x = 3/10, t=1/6, y = √91/10
Now, we know that
2y dy/dt = 1/25 (-9πsin(2πt))
so,
dy/dt = -(9π√3)/(10√91)
As expected, it is falling, if the x end is moving right.
Sure! I'd be happy to help you solve this problem. Let's break it down into parts.
a) To find the time of 1 complete cycle of the rod, we need to determine when the position of the x-axis endpoint repeats itself. In this case, the x-coordinate is given by x(t) = (3/5) sin (πt). A complete cycle occurs when the sine function goes through one full period. The standard period for the sine function is 2π.
To solve for the period of the given function, we need to set the argument of the sine function equal to 2π.
So, (πt) = 2π.
Dividing both sides of the equation by π, we get t = 2.
Therefore, the time of 1 complete cycle of the rod is 2 seconds.
b) To determine the lowest point reached by the end of the rod on the y-axis, we need to find the minimum value of the y-coordinate. In this case, the y-coordinate is obtained by substituting the value of t into the expression y(t) = 0.
Since the y-coordinate is always 0, there is no lowest point reached on the y-axis.
c) To find the speed of the y-axis endpoint when the x-axis endpoint is (3/10, 0), we need to find the derivative of the y-coordinate function with respect to time.
The y-coordinate is given by y(t) = y = 0.
Since the y-coordinate does not depend on time, the speed is equal to 0.
In summary:
a) The time of 1 complete cycle of the rod is 2 seconds.
b) The lowest point reached by the end of the rod on the y-axis is 0.
c) The speed of the y-axis endpoint is 0 when the x-axis endpoint is (3/10, 0).
I hope this helps you solve the problem! Let me know if you have any further questions.