Equate the height to the function given and solve for x
[tex]\begin{gathered} h=50\cos (\frac{\pi}{15}(x-10))+52 \\ \text{where h= 80} \end{gathered}[/tex][tex]\begin{gathered} 80=50\cos (\frac{\pi}{15}(x-10))+52 \\ \text{collect like terms} \\ 80-52=50\cos (\frac{\pi}{15}(x-10)) \\ 28=50\cos (\frac{\pi}{15}(x-10)) \\ \text{Divide both sides by 50} \\ \frac{28}{50}=\cos (\frac{\pi}{15}(x-10)) \end{gathered}[/tex][tex]\begin{gathered} \frac{\pi}{15}(x-10)=\cos ^{-1}(\frac{28}{50}) \\ \frac{\pi}{15}(x-10)=0.9764\text{ or }\frac{\pi}{15}(x-10)=2\pi-0.9764=5.3068 \end{gathered}[/tex][tex]\begin{gathered} x-10=\frac{15}{\pi}(0.9764)\text{ or }x-10=\frac{15}{\pi}(5.3068) \\ x=\frac{15}{\pi}(0.9764)_{}+10\text{ or }x=\frac{15}{\pi}(5.3068)_{}+10 \end{gathered}[/tex][tex]\begin{gathered} x=\text{ 4.7746(0.9764)+10 or x= 4.7746(5.3068)}+10 \\ x=4.662+10\text{ or x=25.338+10} \\ x=14.662\text{ or }x=35.338 \end{gathered}[/tex]Hence, the right options are Option1 and Option5, which are 14.665seconds and 35.338seconds.