Cheryl Peck purchased a computer network for her classroom. The computer network cost $$100,000.$ She estimates that she can charge $$500$ for one session in the classroom. Cheryl knows that enrollment will increase over time. She estimates $50$ students the first year, $75$ students the second year, $100$ students the third year, and $150$ students the fourth year. If her cost of capital is $12$ percent, what is the approximate profitability index?

a$.$

$1\mathrm{.}88$

b$.$

$1\mathrm{.}05$

c$.$

$1\mathrm{.}19$

d$.$

$1\mathrm{.}35$