If we want to receive $\$6000$ per year forever from a continuously compounded account with a $9\%$ annual interest rate, how much should we deposit now? Write your answer as a whole number.

The probability that a machine part fails after $c$ hours of use is $0.00003{\int }_c^{\infty }{e}^{-0.00003t}dt$. Find the probability that the part fails after $25,000$ hours. Round your answer to two decimal places.

Suppose the average waiting time for a certain event is modeled by $0.0002{\int }_0^{\infty }t{e}^{-0.0002t}dt$. What is the average waiting time?

Compute the Laplace transform of $f\left(t\right)=3t$.

A country's water consumption rate is modeled by $w\left(t\right)=w_0{e}^{-qt}$, where $q>0$, $w_0=2\times {10}^8$ $\text{L/yr}$, and the total water reserve is $8\times {10}^{10}$ $\text{L}$. Find the minimum decay constant $q$ so that the water reserve lasts indefinitely.