On a particular day in Salt Lake, UT, the temperature was given by $-t^2 +12t+50$ where $t$ is the time in hours past noon. What is the largest $t$ value at which the temperature was exactly 77 degrees?

We need to solve this

-t^2 + 12t + 50 = 77 subtract 77 from both sides

-t^2 + 12t - 27 = 0 multiply through by -1

t^2 - 12t + 27 = 0

Factor

(t -9) ( t -3) = 0

Set each factor to 0 and solve for t

t - 9 = 0 t - 3 = 0

t =9 t = 3

So....9 hours past noon = 9 PM

Thank you so much! It makes sense how you did it!