Barnaby’s grandfather is always complaining that $50$ years ago, he could buy his girlfriend dinner for only $$1.50$. Now that same dinner costs $$25.25$.

1. Assuming the price has increased exponentially and that same dinner never cost less than $$0.20$, write the equation of a function that models the cost in relationship to the number of years from now.

   Hint (a):

   Begin by defining variables.
   Let $x =$ the number of years
   Let $y =$ the cost of a dinner
   You need to write an equation of the form $y = ab^{x} + k$.

   More Help (a):

   Use the points $\left(−50, 1.50\right)$ and $\left(0, 25.25\right)$. The horizontal asymptote (or vertical shift in this case) is $y = 0.20$ (or $k = 0.20$).

2. How much would you expect the same dinner to cost $60$ years from now?

   Hint (b):

   Let $x = 60$ in your equation from part (a).

   Answer (b):

   Answers vary depending on precision used, about $$874.64.$