After Lee gives his little sister Kara a big push on a swing, her horizontal position as a function of time is given by the equation x(t)= 3cost(e^-0.05t), where x(t) is her horizontal displacement, in metres, from the lowest point of her swing, as a function of time, t, in seconds.
a) From what horizontal distance feom the bottom of Kara's swing did Lee push his sister.
B) Determine the greatest speed Kara will reach and when this occurs.
C) How long will it take for Kara's max horizontal displacement at the top of her swing arc to diminish to 1 m?
To answer these questions, we need to understand the given equation, x(t) = 3cos(t * e^(-0.05t)), which represents Kara's horizontal displacement from the lowest point of her swing as a function of time.
a) To determine from what horizontal distance Lee pushed his sister, we need to find the initial value of x(t) when t is close to 0. We can substitute t = 0 into the equation: x(0) = 3cos(0 * e^(-0.05 * 0)). Since cos(0) = 1, the equation simplifies to x(0) = 3 * 1 * e^(0). Since e^0 is equal to 1, the final answer is x(0) = 3. Therefore, Lee pushed his sister from a horizontal distance of 3 meters from the bottom of Kara's swing.
b) The greatest speed Kara will reach corresponds to the maximum of the absolute value of the derivative of x(t) with respect to t. To find this maximum, we differentiate x(t) with respect to t:
x'(t) = -3e^(-0.05t) * sin(t * e^(-0.05t)) + 0.05 * 3e^(-0.05t) * cos(t * e^(-0.05t))
Now we need to find the critical points by setting x'(t) equal to zero and solving for t. However, given the complexity of the equation, this may not yield exact solutions. Instead, we can use numerical methods to approximate the critical points.
Using the given equation x'(t), we can find the critical points where the derivative equals zero using a numerical method such as the Newton-Raphson method or a graphing calculator. Once we find the values of t at which x'(t) = 0, we can substitute these values back into the original equation x(t) to find the corresponding horizontal displacements x(t).
Next, we can calculate the speed at each critical point by taking the absolute value of x'(t). The greatest speed will correspond to the largest absolute value among these values.
c) To determine how long it will take for Kara's maximum horizontal displacement at the top of her swing arc to diminish to 1 meter, we need to solve the equation x(t) = 1 for t. However, the given equation x(t) = 3cos(t * e^(-0.05t)) cannot be directly solved algebraically for t.
We can use numerical methods, such as the Newton-Raphson method or a graphing calculator, to find an approximate value of t for x(t) = 1. These methods involve iteratively refining an initial guess until it converges to the desired solution. Once we find the value of t at which x(t) = 1, we will have the time it takes for Kara's maximum horizontal displacement to diminish to 1 meter.