The number of visitors P to a website in a given week over a​ 1-year period is given by Upper P left parenthesis t right parenthesis equals 115 plus left parenthesis t minus 85 right parenthesis e Superscript 0.02 tP(t)=115+(t−85)e0.02t​, where t is the week and 1less than or equals≤tless than or equals≤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. The critical point is left parenthesis 35 comma 14.312 right parenthesis(35,14.312).

We are given a function for the number of visitors to a website in a week over a 1-year period:

\[
P(t) = 115 + (t - 85)e^{0.02t}
\]
where \( t \) represents the week and \( 1 \leq t \leq 52 \).

### Part (a) & (b) - Increasing and Decreasing Intervals:
To find where the number of visitors is increasing or decreasing, we need to analyze the first derivative of \( P(t) \), \( P'(t) \). 

#### Step 1: Find the derivative \( P'(t) \)
We use the product and chain rules to differentiate \( P(t) \):
\[
P(t) = 115 + (t - 85)e^{0.02t}
\]
\[
P'(t) = \frac{d}{dt}\left((t - 85)e^{0.02t}\right)
\]
Applying the product rule:
\[
P'(t) = (t - 85) \cdot \frac{d}{dt}\left(e^{0.02t}\right) + e^{0.02t} \cdot \frac{d}{dt}(t - 85)
\]
Since \(\frac{d}{dt}\left(e^{0.02t}\right) = 0.02e^{0.02t}\) and \(\frac{d}{dt}(t - 85) = 1\), we get:
\[
P'(t) = (t - 85) \cdot 0.02e^{0.02t} + e^{0.02t}
\]
Factor out \( e^{0.02t} \):
\[
P'(t) = e^{0.02t}\left(0.02(t - 85) + 1\right)
\]
Simplify inside the parentheses:
\[
P'(t) = e^{0.02t}(0.02t - 1.7 + 1)
\]
\[
P'(t) = e^{0.02t}(0.02t - 0.7)
\]

#### Step 2: Analyze the sign of \( P'(t) \)
We now solve for when \( P'(t) = 0 \):
\[
e^{0.02t}(0.02t - 0.7) = 0
\]
Since \( e^{0.02t} \neq 0 \) for any \( t \), we solve:
\[
0.02t - 0.7 = 0
\]
\[
0.02t = 0.7
\]
\[
t = \frac{0.7}{0.02} = 35
\]

Thus, the critical point occurs at \( t = 35 \).

#### Step 3: Determine the intervals of increase and decrease
- For \( t < 35 \), \( 0.02t - 0.7 < 0 \), so \( P'(t) < 0 \) and the number of visitors is **decreasing**.
- For \( t > 35 \), \( 0.02t - 0.7 > 0 \), so \( P'(t) > 0 \) and the number of visitors is **increasing**.

### Conclusion for parts (a) and (b):
- The number of visitors is **decreasing** on the interval \( (1, 35) \).
- The number of visitors is **increasing** on the interval \( (35, 52) \).

### Part (c) - Critical Point:
The critical point is given as \( (35, 14.312) \). This means:
- At week \( t = 35 \), the number of visitors reaches a minimum of approximately 14.312.
- This critical point signifies the week during the year where the visitor count stops decreasing and starts increasing again.

### Follow-up questions:
1. How does the behavior of \( P(t) \) change as \( t \) approaches the endpoints \( t = 1 \) and \( t = 52 \)?
2. What happens to the number of visitors as \( t \to \infty \), based on the nature of the exponential term?
3. Can you find the total number of visitors throughout the year by integrating \( P(t) \)?
4. How sensitive is the function to changes in \( t \), particularly near the critical point \( t = 35 \)?
5. How would the intervals change if the growth rate in the exponential term was larger or smaller?

**Tip**: Critical points occur where the derivative equals zero or is undefined. Always check these points for potential maxima, minima, or points of inflection.

The number of visitors P to a website in a given week over a 1-year period is given by P(t) = 115 + (t − 85)e^(0.02t), where t is the week and 1 ≤ t ≤ 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. The critical point is (35, 14.312).

Discover how to analyze a function representing website visitors with P(t) = 115 + (t − 85)e^(0.02t). Learn where visitor numbers decrease, increase, and find the critical point at (35, 14.312). This problem explores calculus concepts such as derivatives, exponential functions, and critical points. Suitable for advanced high school and college students.

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