Question

1. The exponential function modeled below represents the number of squarekilometers of land occupied by cane toads x years after this animal was firstintroduced into Australia,Area Occupied by Cane ToadsTimeArea1,200(y)(km)1,1001,000036,500900553,6008007001078,80060015115,78050020 170,12040025250,00030020030 367,300100539,7000 5 10 15 20 25 30 35Time (y)Based on the data, which measurement is closest to the number of squarekilometers of land that will be occupied by cane toads 40 years after thisArea (thousands of km)35< PREVIOUS

Answer 1

Step 1: Write out the formula

$\begin{gathered} f(t)=a(1+r)^t \\ \text{where} \\ f(x)=\text{ area occupied by the cane toad} \\ a=\text{ initial amount} \\ r=\text{ growth rate} \\ t=\text{ number of years} \end{gathered}$

Step 2: Substitute the given values

From the table, we can see that

$f(0)=a=36500\operatorname{km}^2$$\begin{gathered} f(5)=a(1+r)^5=53600 \\ \text{ This implies that} \\ 36500(1+r)^5=53600 \\ \text{thus} \\ (1+r)^5=\frac{53600}{36500} \\ \text{Therefore} \\ (1+r)^5=\frac{536}{365} \\ \text{thus} \\ 1+r=\sqrt[5]{\frac{536}{365}}\approx1.0799 \end{gathered}$$\begin{gathered} \text{Hence,} \\ r=1.0799-1=0.0799 \end{gathered}$

Therefore,

$f(t)=36500(1.0799)^t$

Step 3: Substitute time t = 40 into f(t)

$f(40)=36500(1.0799)^{40}=790013\approx800000$

Therefore, based on the data, the measurement that is closest to the number of square kilometers of land that will be occupied by cane toads 40 years after this is 800000