Question

after t seconds, the height of a weight attached to a spring is s(t)= -3 cos 2$\pi$t inches. when is the first time that the weight reaches its maximum height?

Answer 1

so hmm notice the picture below

so it reaches its maximum when y = 0

thus $\bf s(t)=-3cos(2\pi t)\implies 0=-3cos(2\pi t)\implies 0=cos(2\pi t) \\\\\\ cos^{-1}(0)=cos^{-1}[cos(2\pi t)]\implies cos^{-1}(0)=2\pi t \\\\\\ \cfrac{cos^{-1}(0)}{2\pi }=t \\\\\\ \textit{now, on the interval of }[0,2\pi ]\ where\ cos^{-1}(0)? \\\\\\ well\qquad \frac{\pi }{2}\ ,\ \frac{3\pi }{2}\qquad thus \\\\\\ t= \begin{cases} \cfrac{\frac{\pi }{2}}{2\pi }\implies \cfrac{1}{4}\\\\ \cfrac{\frac{3\pi }{2}}{2\pi }\implies \cfrac{3}{4} \end{cases}$