Question

The number of houses in Central Village, New York, grows every year according to the function H(t) = 540(1.039)^t . where H represents the number of houses, and i represents the number of years since January 1995. A civil engineering firm has suggested that a new, larger well must be built by the village to supply its water when the number of houses exceeds 1.000. During which year will this first happen?

Answer 1

`\begin{gathered} We\text{ have to find t such that } \\ H(t)=1000 \\ 540(1.039)^t=1000 \\ (1.039)^t=(1000)/(540) \\ (1.039)^t=1.851851851851 \\ \ln (1.039^t)=\ln (1.851851851851) \\ t\ln (1.039)=\ln (1.851851851851) \\ t=(\ln (1.851851851851))/(\ln (1.039)) \\ t=16.1058 \\ \\ \text{Approximately after 16 years, that is on 1995+16=}2011 \\ \text{During 2011} \end{gathered}`