The number of visitors $\mathrm{P}$ to a website in a given week over a 1-year period is given by $\mathrm{P}(\mathrm{t})=119+(\mathrm{t}-82)e^{0.02t}$, where $t$ is the week and $1\le t\le 52$.
a) Over what interval of time during the 1 -year period is the number of visitors decreasing?
b) Over what interval of time during the 1 -year period is the number of visitors increasing?
c) Find the critical point, and interpret its meaning.
a) The number of visitors is decreasing over the interval
(Simplify your answer. Type integers or decimals rounded to three decimal places as needed. Type your answer in interval notation.)

Step 1 ：Given the function $P(t)=119+(t-82)e^{0.02t}$, where $t$ is the week and $1\le t\le 52$.

Step 2 ：To find out when the number of visitors is decreasing, we need to find the derivative of the function $P(t)$.

Step 3 ：The derivative of $P(t)$ is $P^{\prime }(t)=0.02(t-82)e^{0.02t}+e^{0.02t}$.

Step 4 ：We set the derivative equal to zero to find the critical points: $0.02(t-82)e^{0.02t}+e^{0.02t}=0$.

Step 5 ：Solving this equation gives us the critical point $t=32$.

Step 6 ：We then test the intervals around this critical point to see where the derivative is negative, which indicates that the function is decreasing.

Step 7 ：The intervals are $(1,32)$ and $(32,52)$.

Step 8 ：Testing these intervals, we find that the derivative is negative in the interval $(1,32)$, indicating that the number of visitors is decreasing during this time.

Step 9 ：Final Answer: The number of visitors is decreasing over the interval $\boxed{{\displaystyle (1,32)}}$.