2 A breeding program to ensure the survival of pygmy possums is established with an initial population of 50 ( 25 pairs). From a previous program, the expected population $$ P $$ in $$ n $$ years' time is given by $$ P(n)=P_{0} imes(1.23)^{n} $$. a What is the value of $$ P_{0} $$ ? b What is the expected population after: I 2 years ii 5 years iii 10 years? c Sketch the graph of the population over time using $$ \mathbf{a} $$ and $$ \mathbf{b} $$ only. d Hence estimate the time needed for the population to reach 500 . e Use technology to graph $$ \mathrm{Y}_{1}=50 imes(1.23)^{\mathrm{x}} $$. Hence check your answers to d .

Question

2 A breeding program to ensure the survival of pygmy possums is established with an initial population of 50 ( 25 pairs). From a previous program, the expected population $$ P $$ in $$ n $$ years' time is given by $$ P(n)=P_{0} 	imes(1.23)^{n} $$. a What is the value of $$ P_{0} $$ ? b What is the expected population after: I 2 years ii 5 years iii 10 years? c Sketch the graph of the population over time using $$ \mathbf{a} $$ and $$ \mathbf{b} $$ only. d Hence estimate the time needed for the population to reach 500 . e Use technology to graph $$ \mathrm{Y}_{1}=50 	imes(1.23)^{\mathrm{x}} $$. Hence check your answers to d .

UpStudy AI · Accepted Answer

**a. Calculation of $$ P_{0} $$:**

Since the initial population is given as 50, we have  
$$
P_{0} = 50.
$$

---

**b. Expected population after a given number of years:**

The population after $$ n $$ years is given by  
$$
P(n) = 50 \times (1.23)^n.
$$

**i. After 2 years ($$ n=2 $$):**

$$
P(2) = 50 \times (1.23)^2.
$$

First, calculate $$(1.23)^2$$:  
$$
(1.23)^2 = 1.5129.
$$

Then,  
$$
P(2) \approx 50 \times 1.5129 = 75.645.
$$

So, after 2 years, the expected population is approximately 76 possums (rounding to the nearest whole number).

---

**ii. After 5 years ($$ n=5 $$):**

$$
P(5) = 50 \times (1.23)^5.
$$

We first calculate $$(1.23)^5$$. A rough calculation gives:  
$$
(1.23)^5 \approx 2.815.
$$

Then,  
$$
P(5) \approx 50 \times 2.815 = 140.75.
$$

So, after 5 years, the expected population is approximately 141 possums.

---

**iii. After 10 years ($$ n=10 $$):**

$$
P(10) = 50 \times (1.23)^{10}.
$$

Since $$(1.23)^{10}$$ is the square of $$(1.23)^5$$, we have:  
$$
(1.23)^{10} \approx (2.815)^2 \approx 7.929.
$$

Then,  
$$
P(10) \approx 50 \times 7.929 = 396.45.
$$

Thus, after 10 years, the expected population is approximately 396 possums.

---

**c. Sketching the graph of the population over time:**

Plot the function  
$$
P(n)=50 \times(1.23)^n
$$
using the points calculated above:  
- When $$ n=0 $$, $$ P(0)=50 $$.  
- When $$ n=2 $$, $$ P(2)\approx75.65 $$.  
- When $$ n=5 $$, $$ P(5)\approx140.75 $$.  
- When $$ n=10 $$, $$ P(10)\approx396.45 $$.

The graph will be an exponentially increasing curve starting at 50 and rising more steeply as $$ n $$ increases.

---

**d. Estimating the time needed for the population to reach 500:**

We need to solve the equation  
$$
50 \times (1.23)^n = 500.
$$

Divide both sides by 50:  
$$
(1.23)^n = 10.
$$

Take the natural logarithm of both sides:  
$$
\ln((1.23)^n) = \ln(10).
$$

Using the logarithm property, $$\ln((1.23)^n) = n \cdot \ln(1.23)$$, we have:  
$$
n \cdot \ln(1.23) = \ln(10).
$$

Solve for $$ n $$:  
$$
n = \frac{\ln(10)}{\ln(1.23)}.
$$

Using approximate values, $$\ln(10) \approx 2.302585$$ and $$\ln(1.23) \approx 0.207014$$:  
$$
n \approx \frac{2.302585}{0.207014} \approx 11.12.
$$

So, it will take approximately 11 years for the population to reach 500.

---

**e. Graphing $$ Y_{1}=50 \times (1.23)^x $$ using technology:**

When you use graphing technology (such as a graphing calculator or software) to plot  
$$
Y_{1}=50 \times (1.23)^x,
$$  
you will see an exponential curve that starts at 50 when $$ x=0 $$ and rises rapidly as $$ x $$ increases. By tracing the curve, you can check that it crosses the horizontal line $$ Y=500 $$ at about $$ x \approx 11 $$ years. This confirms our earlier calculation in part (d).

**a. Initial Population ($$ P_{0} $$):** 50 possums. **b. Expected Population After:** - **i. 2 Years:** Approximately 76 possums. - **ii. 5 Years:** Approximately 141 possums. - **iii. 10 Years:** Approximately 396 possums. **c. Graph of Population Over Time:** An exponentially increasing curve starting at 50 possums and rising steeply as time increases. **d. Time to Reach 500 Possums:** Approximately 11 years. **e. Technology Check:** Using graphing technology to plot $$ Y_{1}=50 \times (1.23)^x $$ confirms that the population reaches 500 at around 11 years.

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