If I know the weekly sales of calculators can be modeled by the equation s(t)=10-te^(-t/20) where s is the number of calculators sold per week after t weeks, how many calculators will be sold in be first 60 weeks?

that would be the sum of the first 60 weeks' sales. In other words,

∫[0,60] (10 - te^(-t/20)) dt ≈ 280