The probability that a certain machine breaks during the first $370$ hours of its use is $50\%$. How many hours may the machine be used until it breaks with a probability, which at least $85\%$?
I don't know if I solved this right. In my equation, $t$ is the answer to the question?
$$\frac{1-e^{- \lambda .370}}{1-e^{- \lambda (370+t)}} = \frac{0.5}{0.85}$$
You are correct in using the exponential distribution. The first hint is there to tell you how to calculate $\lambda$:
$$1-e^{-\lambda 370}=0.5\implies \lambda = -\frac{\ln(2)}{370}$$
Now, use this lambda to solve for your desired time. Can you solve this part?