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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

User Ora
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1 Answer

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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=(53600)/(36500) \\ \text{Therefore} \\ (1+r)^5=(536)/(365) \\ \text{thus} \\ 1+r=\sqrt[5]{(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

User Krystan Honour
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