Question

the population of fish in a certain lake at time t months is given by the function p(t) = 21,000 / 1 + 24 (2^-0.34t). use a graphing calculator or other technology to complete parts (a) through (d) below. (a) graph the population function from t = 0 to t = 48 (a four - year period). choose the correct graph below. oa. ob. oc. od.

Explanation:

Step1: Analyze the function type

The population function is \( p(t)=\frac{21000}{1 + 24(2^{-0.34t})} \), which is a logistic growth function. For a logistic function \( \frac{L}{1 + e^{-kt}} \) (or similar forms), as \( t \to 0 \), we can find the initial value: when \( t = 0 \), \( 2^{-0}=1 \), so \( p(0)=\frac{21000}{1 + 24\times1}=\frac{21000}{25} = 840 \). As \( t \to \infty \), \( 2^{-0.34t}\to0 \), so \( p(t)\to\frac{21000}{1}=21000 \) (wait, the graph's y - axis is up to 25000, but the limit is 21000? Wait, maybe I miscalculated. Wait, the function is \( p(t)=\frac{21000}{1 + 24(2^{-0.34t})} \). When \( t = 0 \), denominator is \( 1+24 = 25 \), so \( p(0)=21000/25 = 840 \). As \( t \) increases, \( 2^{-0.34t} \) decreases, so denominator decreases, so \( p(t) \) increases towards 21000. Now let's check the graphs:

- Option A: The y - axis is labeled with 25000, but the curve starts from a high value and decreases? No, population should grow (since the exponent of 2 is negative, so \( 2^{-0.34t} \) decreases, denominator decreases, so \( p(t) \) increases). So A is decreasing, wrong.
- Option B: Let's see the shape. The curve starts at 0? No, our initial value is 840, not 0. Wait, maybe the graph's y - axis is p(t) and x - axis is t (0 to 48). Wait, maybe I misread the function. Wait, the function is \( p(t)=\frac{21000}{1 + 24(2^{-0.34t})} \). Wait, maybe the numerator is 25000? No, the problem says 21000. Wait, maybe the graph's y - axis is up to 25000, and the function approaches 21000. Let's check the behavior: as t increases, \( 2^{-0.34t} \) gets smaller, so \( 1 + 24(2^{-0.34t}) \) gets closer to 1, so \( p(t) \) gets closer to 21000. At t = 0, p(0)=21000/(1 + 24)=840. So the curve should start at a low value (around 800 - 900) and increase towards 21000. Let's check the options:

- Option B: The curve starts at 0, which is wrong.
- Option C: Let's see the shape. The curve starts at 0? No, initial value is 840. Wait, maybe the graph's scaling is different. Wait, the x - axis is t from 0 to 48, y - axis p(t) from 0 to 25000. Let's check the function's derivative (rate of change). The logistic function has an S - shape: starts with slow growth, then faster, then slow again as it approaches the carrying capacity. Wait, our function: let's rewrite \( 2^{-0.34t}=e^{-0.34t\ln2}\approx e^{-0.235t} \). So the function is \( p(t)=\frac{21000}{1 + 24e^{-0.235t}} \), which is a logistic growth function with carrying capacity \( L = 21000 \), initial population \( p(0)=21000/25 = 840 \), and growth rate related to 0.235. So the graph should start at a low value (around 800), increase, and approach 21000. Now looking at the options:

- Option C: The curve starts at 0? No, but maybe the graph's y - axis is p(t) and the initial value is close to 0? Wait, no, 840 is not 0. Wait, maybe I made a mistake in the function. Wait, the problem says "the population of fish in a certain lake at time t months is given by the function \( p(t)=\frac{21000}{1 + 24(2^{-0.34t})} \)". Wait, maybe the numerator is 25000? No, the user's image shows the function as \( p(t)=\frac{21,000}{1 + 24(2^{-0.34t})} \). Wait, let's check t = 48: \( 2^{-0.34\times48}=2^{-16.32}\approx2^{-16}=1/65536\approx0 \), so \( p(48)\approx21000/1 = 21000 \). So the graph should approach 21000 as t approaches 48. Now let's check the options:

- Option B: The curve is increasing, and as t approaches 48, it's close to 25000? No, our limit is 21000. Wait, maybe the function is \( p(t)=\frac{25000}{1 + 24(2^{-0.34t})} \)? Maybe a typo, but assuming the graph's y - axis is 25000, and the carrying capacity is 25000. Then at t = 0, \( p(0)=25000/25 = 1000 \), and as t approaches 48, \( p(48)\approx25000 \). Then the logistic curve starts at 1000, increases, and approaches 25000. Now looking at the options:

- Option C: The curve has an S - shape, starts at 0? No. Option B: The curve starts at 0, then increases, and at t = 48, it's close to 25000? Wait, no, the x - axis is t (0 to 48), y - axis p(t) (0 to 25000). Let's think about the shape of a logistic curve: it starts with slow growth, then rapid growth, then slow growth as it approaches the carrying capacity. So the curve should be concave up then concave down. Now looking at the options:

- Option C: The curve starts at 0, increases, and the shape is S - like. Wait, maybe the initial value is close to 0? No, our initial value is 840 (or 1000 if numerator is 25000). Wait, maybe the correct graph is Option C? Wait, no, let's re - evaluate. The function is a logistic growth function, so it should have an S - curve, starting from a low value, increasing, and approaching the carrying capacity. Among the options, Option C's curve has the S - shape, starting from near 0 (maybe the graph's scaling makes the initial value look like 0) and increasing towards 25000. Wait, maybe I was wrong about the initial value. Wait, if the numerator is 25000, then \( p(0)=25000/25 = 1000 \), which is close to 0 on a graph with y - axis up to 25000. So the correct graph should be the one with the S - curve, starting from a low value, increasing, and approaching 25000. So Option C? Wait, no, let's check the options again:

- Option A: Decreasing, wrong.
- Option B: The curve is increasing, but the shape is more like it's increasing rapidly at the end. Wait, logistic curve has the inflection point where the growth rate changes. The inflection point for \( p(t)=\frac{L}{1 + ae^{-kt}} \) is at \( t=\frac{\ln a}{k} \). Here, \( a = 24 \), \( k = 0.34\ln2\approx0.235 \), so \( t=\frac{\ln24}{0.235}\approx\frac{3.178}{0.235}\approx13.5 \). So around t = 13 - 14, the growth rate changes. So the curve should be concave up then concave down. Looking at the options, Option C has a curve that starts, increases, and the shape is S - like. Option B's curve seems to be concave down then concave up? No, logistic is concave up then concave down. So Option C? Wait, maybe the correct answer is Option C. Wait, no, let's check the x - axis and y - axis labels. Option C: x - axis t (0 - 48), y - axis p(t) (0 - 25000), curve starts at 0, increases, and has the S - shape. So I think the correct graph is Option C.

Answer:

C