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

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 (\pi )/(2)\ ,\ (3\pi )/(2)\qquad thus \\\\\\ t= \begin{cases} \cfrac{(\pi )/(2)}{2\pi }\implies \cfrac{1}{4}\\\\ \cfrac{(3\pi )/(2)}{2\pi }\implies \cfrac{3}{4} \end{cases}`