91Ó°ÊÓ

To understand the direction of the pulse, it's important to notice that the term affecting the x displacement is (4x + 5t). To see the effect of time on the amplitude, we will look at how the function behaves with increasing t values. If Y(x, t) increases with the time, it's moving in the positive x-direction; otherwise, it's moving in the negative x-direction. We need to find different values of Y(x, t) for different consecutive time values, keeping x constant. Let x = 0, t = 1 and t = 2, we'll find Y(0, 1) and Y(0, 2) \(Y(0, 1) = \frac{0.8}{(4 \cdot 0 + 5 \cdot 1)^2 + 5} = \frac{0.8}{30}\) and \(Y(0, 2) = \frac{0.8}{(4 \cdot 0 + 5 \cdot 2)^2 + 5} = \frac{0.8}{65}\) Since \(Y(0, 2) < Y(0, 1)\), the pulse is moving in the positive x-direction. Hence, option (A) is correct.

In order to find the distance traveled by the pulse in 2 seconds, we need to find the difference in the x-positions where the amplitude is maximum at t = 0 and t = 2. To do this, we need to take the derivative of Y(x, t) with respect to x and find its maximum point: \(\frac{d Y(x, t)} {d x} = \frac{-12.8 (4x + 5t)}{[(4x + 5t)^{2} + 5]^{2}}\) Since Y(x, t) is maximum, we can set the derivative to zero, and solve for x: \(\frac{-12.8 (4x + 5t)}{[(4x + 5t)^{2} + 5]^{2}} = 0\) Solving for x when t = 0 gives \(4x = 0 \Rightarrow x_0 = 0\) and when t = 2, gives \(4x + 5 \cdot 2 = 0 \Rightarrow x_2 = -2.5\) Now, find the distance traveled by the pulse in 2 seconds: \(|x_2 - x_0| = |-2.5 - 0| = 2.5\) Thus, in 2 seconds, the pulse travels a distance of 2.5 meters. Option (B) is correct.

To find the maximum displacement, we need to find the maximum value of Y(x, t). We already determined that the maximum value for Y(x, t) occurs at x = 0 and t = 0. Calculate the value of Y(x, t) at x = 0 and t = 0: \(Y(0, 0) = \frac{0.8}{(4 \cdot 0 + 5 \cdot 0)^{2} + 5} = \frac{0.8}{5}\) Thus, the maximum displacement of the pulse is 0.16 meters. Option (C) is correct.

In order to determine if the pulse is symmetric at t = 0, we need to check if it satisfies the condition: \(Y(x, 0) = Y(-x, 0)\) Check the pulse's symmetry at t = 0: \(Y(x, 0) = \frac{0.8}{(4x)^{2} + 5} \) and \(Y(-x, 0) = \frac{0.8}{(4(-x))^{2} + 5} = \frac{0.8}{(4x)^{2} + 5}\) Since \(Y(x, 0) = Y(-x, 0)\), the pulse is symmetric at t = 0. Hence, option (D) is correct.