{Use of Tech} Cell population The ​population ​of a culture of cells after t days is approximated by the function P(t)=1600 / 1 + 7e^−0.02t, for t≥0.
a. Graph the population function.  

Verified step by step guidance

1. Step 1: Understand the function P(t) = \frac{1600}{1 + 7e^{-0.02t}}. This is a logistic function, which is commonly used to model population growth where the growth rate decreases as the population reaches a carrying capacity.
2. Step 2: Identify key features of the logistic function. The carrying capacity, or the maximum population, is 1600. The initial population can be found by evaluating P(0).
3. Step 3: Determine the horizontal asymptotes. As t approaches infinity, the term 7e^{-0.02t} approaches zero, so the function approaches the carrying capacity of 1600. As t approaches negative infinity, the function approaches zero.
4. Step 4: Calculate a few key points to help sketch the graph. Evaluate P(t) at t = 0, t = 10, t = 20, etc., to see how the population changes over time.
5. Step 5: Use a graphing tool or software to plot the function P(t) = \frac{1600}{1 + 7e^{-0.02t}}. Observe the S-shaped curve typical of logistic growth, starting near zero, increasing rapidly, and then leveling off near the carrying capacity.