Question

Letx(t) = t - sin(t) and y(t) = 1 - cos(t); Att = 6

Answer 1

Solution:

Given;

$\begin{gathered} x(t)=t-\sin(t) \\ \\ y(t)=1-\cos(t) \\ \\ t=6 \end{gathered}$$\begin{gathered} x(6)=6-\sin(6) \\ \\ x(6)=5.90 \end{gathered}$$\begin{gathered} y(6)=1-\cos(6) \\ \\ y(6)=0.01 \end{gathered}$

Then;

$\begin{gathered} \frac{dx}{dt}=1-\cos(t) \\ \\ t=6; \\ \\ \frac{dx}{dt}|_{t=6}=1-\cos(6) \\ \\ \frac{dx}{dt}|_{t=6}=0.01 \end{gathered}$$\begin{gathered} \frac{dy}{dt}=\sin(t) \\ \\ t=6; \\ \\ \frac{dy}{dt}|_{t=6}=\sin6 \\ \\ \frac{dy}{dt}|_{t=6}=0.10 \end{gathered}$$\begin{gathered} \frac{dy}{dx}=\frac{dy}{dt}\div\frac{dx}{dt} \\ \\ \frac{dy}{dx}=\frac{\sin(t)}{1-\cos(t)} \\ \\ \frac{dy}{dx}|_{t=6}=\frac{\sin(6)}{1-\cos(6)} \\ \\ \frac{dy}{dx}|_{t=6}=19.08 \end{gathered}$$\begin{gathered} speed=\sqrt{(1-\cos t)^2+(\sin t)^2} \\ \\ speed=\sqrt{1-2\cos t+\cos^2t+\sin^2t} \\ \\ speed=\sqrt{2-2\cos t} \\ \\ t=6; \\ \\ speed=\sqrt{2-2\cos(6)} \\ \\ speed=0.10 \end{gathered}$