The credit remaining on a phone card (in dollars) is a linear function of the total calling time made with the card (in minutes). The remaining credit after 37 minutes of calls is $14.82, and the remaining credit after 70 minutes of calls is $10.20. What is the remaining credit after 100 minutes of calls?
Let $m$ be the remaining credit after $100$ minutes of calls. We are given that $m$ is a linear function of time: $m=pt+b$ for some constants $p$ and $b.$ The remaining credit after $37$ minutes of calls is $14.82$ dollars, so $14.82=37p+b.$ The remaining credit after $70$ minutes of calls is $10.20$ dollars, so $10.20=70p+b.$ Substituting the second equation into the first to eliminate $b,$ we get $14.82=37p+(10.20-70p),$ or $4p=4.62.$ Thus, $p=1.155,$ so $m=100(1.155)+b=115.50+b.$ Since $m$ is the remaining credit, we have \[m=\boxed{\$ 115.50}.\]
