Question

this exercise uses the population growth model.
a grey squirrel population was introduced in a certain county of great britain 40 years ago. biologists observe that the population doubles every 8 years, and now the population is 120,000.
(a) what was the initial size of the squirrel population?
______ squirrels.
(b) estimate the squirrel population 10 years from now. (round your answer to the nearest whole number.)
______ squirrels
(c) sketch a graph of the squirrel population. (assume t = 0 corresponds to the initial introduction.)
four graphs of n(t) vs t are shown, with n(t) on the y - axis (ranging from 100000 to 400000) and t on the x - axis (ranging from 0 to 50). the first graph has a curve starting around (0,100000) and increasing; the second starts near (0,0) and increases; the third and fourth are partially visible.

Explanation:

Step1: Recall the population growth model

The formula for exponential population growth is \( n(t) = n_0 \cdot 2^{\frac{t}{T}} \), where \( n(t) \) is the population at time \( t \), \( n_0 \) is the initial population, \( T \) is the time it takes for the population to double.

Step2: Solve part (a)

We know that \( t = 40 \) years, \( n(40)=120000 \), and \( T = 8 \) years. Substitute these values into the formula:
\( 120000 = n_0 \cdot 2^{\frac{40}{8}} \)
\( 120000 = n_0 \cdot 2^{5} \)
\( 120000 = n_0 \cdot 32 \)
Then, solve for \( n_0 \): \( n_0=\frac{120000}{32} = 3750 \)

Step3: Solve part (b)

First, find the population 10 years from now. The current time is \( t = 40 \), so 10 years from now, \( t = 40 + 10=50 \) years.
Using the formula \( n(t) = n_0 \cdot 2^{\frac{t}{T}} \), with \( n_0 = 3750 \), \( T = 8 \), and \( t = 50 \):
\( n(50)=3750\cdot 2^{\frac{50}{8}} \)
\( \frac{50}{8}=6.25 \)
\( n(50)=3750\cdot 2^{6.25} \)
\( 2^{6.25}=2^{6 + 0.25}=2^6\cdot 2^{0.25}=64\cdot\sqrt[4]{2}\approx64\times1.1892 = 76.1088 \)
\( n(50)\approx3750\times76.1088\approx285408 \)

Step4: Analyze part (c)

The function \( n(t)=3750\cdot 2^{\frac{t}{8}} \) is an exponential growth function. At \( t = 0 \), \( n(0)=3750 \). As \( t \) increases, \( n(t) \) increases exponentially. The first graph (top - left) has an initial value around 100000, which is incorrect. The second graph (top - right) has an initial value near 0, which is incorrect. The third and fourth graphs: the correct graph should start at \( t = 0 \), \( n(0)=3750 \), and increase exponentially. Wait, maybe I misread the graphs. Wait, in part (a), we found \( n_0 = 3750 \), so at \( t = 0 \), \( n(0)=3750 \). The first graph (top - left) at \( t = 0 \) is around 100000, no. The second graph (top - right) at \( t = 0 \) is 0, no. Wait, maybe the graphs are labeled differently. Wait, the function is \( n(t)=3750\cdot 2^{\frac{t}{8}} \). Let's check at \( t = 40 \), \( n(40)=120000 \). Let's see the first graph: at \( t = 40 \), the value is around 120000? Wait, the first graph (top - left) at \( t = 40 \) (x - axis up to 50, so \( t = 40 \) is near the end) has \( n(t) \) around 120000? Wait, maybe I made a mistake in graph analysis. Wait, the correct graph should be an exponential curve starting at \( (0, 3750) \) and passing through \( (40, 120000) \). Let's check the first graph (top - left): at \( t = 0 \), \( n(t)\approx100000 \), no. The second graph (top - right): at \( t = 0 \), \( n(t)=0 \), no. Wait, maybe the graphs are presented with \( t \) on the x - axis. Wait, perhaps the first graph (top - left) is incorrect, the second (top - right) is incorrect. Wait, maybe I messed up. Wait, the function is \( n(t)=3750\cdot 2^{\frac{t}{8}} \). Let's compute \( n(10)=3750\cdot 2^{\frac{10}{8}}=3750\cdot 2^{1.25}\approx3750\times2.3784 = 8925 \), \( n(20)=3750\cdot 2^{\frac{20}{8}}=3750\cdot 2^{2.5}=3750\times5.6568 = 21213 \), \( n(30)=3750\cdot 2^{\frac{30}{8}}=3750\cdot 2^{3.75}\approx3750\times13.454 = 50452.5 \), \( n(40)=120000 \). So the graph should start low at \( t = 0 \) (3750) and curve upwards, passing through these points. The first graph (top - left) starts at around 100000, which is wrong. The second graph (top - right) starts at 0, wrong. Wait, maybe the graphs are labeled with \( n(t) \) on the y - axis. Wait, the first graph (top - left) has \( n(t) \) starting at ~100000, the second (top - right) at 0. The correct graph should start at 3750, so maybe the first graph is incorrect, the second is incorrect. Wait, perhaps the user made a typo in the graph description. But based on the function, the correct graph is an exponential curve with \( n(0)=3750 \) and increasing.

Answers:

(a) \(\boxed{3750}\)
(b) \(\boxed{285408}\) (rounded to nearest whole number)
(c) The correct graph should be the one that starts at \( t = 0 \) with \( n(0)=3750 \) and shows exponential growth, passing through \( (40, 120000) \). (Assuming the first graph (top - left) was mis - interpreted earlier, but based on the function, the graph with initial value around 3750 and exponential growth. If we re - check, maybe the first graph (top - left) at \( t = 0 \) is around 100000, no. Wait, maybe I made a mistake in part (a). Wait, no, \( 2^{5}=32 \), \( 120000\div32 = 3750 \) is correct. So the initial population is 3750, so the graph should start at (0, 3750). So among the given graphs, the one that starts low (around 3750) and increases exponentially. If the first graph (top - left) has \( n(0)\approx100000 \), no. The second (top - right) has \( n(0)=0 \), no. Maybe the third or fourth, but the description is a bit unclear. But based on the function, the correct graph is an exponential curve with \( n(0) = 3750 \) and \( n(40)=120000 \).

Answer:

s:
(a) \(\boxed{3750}\)
(b) \(\boxed{285408}\) (rounded to nearest whole number)
(c) The correct graph should be the one that starts at \( t = 0 \) with \( n(0)=3750 \) and shows exponential growth, passing through \( (40, 120000) \). (Assuming the first graph (top - left) was mis - interpreted earlier, but based on the function, the graph with initial value around 3750 and exponential growth. If we re - check, maybe the first graph (top - left) at \( t = 0 \) is around 100000, no. Wait, maybe I made a mistake in part (a). Wait, no, \( 2^{5}=32 \), \( 120000\div32 = 3750 \) is correct. So the initial population is 3750, so the graph should start at (0, 3750). So among the given graphs, the one that starts low (around 3750) and increases exponentially. If the first graph (top - left) has \( n(0)\approx100000 \), no. The second (top - right) has \( n(0)=0 \), no. Maybe the third or fourth, but the description is a bit unclear. But based on the function, the correct graph is an exponential curve with \( n(0) = 3750 \) and \( n(40)=120000 \).