How to Solve for t and C in S'(t)=-420t^2/5 Annual sales of a certain cell phone has been declining at a rate of S′(t) = −420t2/5 phones per year, where S(t) is the number of cell phones sold in t years. The company is planning on withdrawing this model when annual sales reach 60,000 cell phones. If current annual sales of these phones are 100,000, find S(t). How long will the company continue to manufacture this phone?
I started by integrating S'(t) = -300t^7/5 + C
But this is where im stuck. Since im not given any indication of the starting t when s(t) = 100000 nor s(t) = 60000, how would I go about solving this equation.
Should I start by finding dS/dt? but yet again I don't have any indication for that.
any pointers would be appreciated. thank you.


You got that $S(t)=-300 t^{7/5}+C.$ Since you know the current $S$ you can assume that now is when $t=0.$ That is: $S(t)=100000-300 t^{7/5}.$
How long will the company continue to manufacture this phone? We have to solve $S(t)=60000.$ In other words: $$100000-300 t^{7/5}=60000.$$
