Rangers at a nature reserve are monitoring the spread of an invasive weed. At the start of monitoring there are 100 weeds. The number of weeds is growing at a rate of 50% per month.
Let \(N\) = number of weeds, and \(t\) = time in months.
- Complete the table of values below that models the growth of the weeds over an 8 month period. Round all answers to the nearest whole number. (2 marks)
\(\begin{array}{|c|c|c|c|c|c|} \hline \vphantom{\dfrac{1}{1}}\quad t \quad & \quad 0 \quad & \quad 1 \quad & \quad 2 \quad & \quad 4 \quad & \quad 8 \quad \\[6pt] \hline \vphantom{\dfrac{1}{1}}N & 100 & & & & \\[12pt] \hline \end{array}\)
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- Using the table of values from (a), neatly plot the points and join with a smooth curve. (2 marks)
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a. \(\text{Table of values:}\)
\begin{array}{|c|c|c|c|c|c|} \hline \quad t \quad & \quad 0 \quad & \quad 1 \quad & \quad 2 \quad & \quad 4 \quad & \quad 8 \quad \\[6pt] \hline \ N & 100 & 150 & 225 & 506 & 2563 \\[6pt] \hline \end{array}
b.
a. \(\text{Table of values:}\)
\begin{array}{|c|c|c|c|c|c|} \hline \quad t \quad & \quad 0 \quad & \quad 1 \quad & \quad 2 \quad & \quad 4 \quad & \quad 8 \quad \\[6pt] \hline \ N & 100 & 150 & 225 & 506 & 2563 \\[6pt] \hline \end{array}
\(\text{Algebraic method}\)
\(\text{Formula: }\ N=100\times1.5^t\)
\(t=0:\ N=100\times1.5^{0}=100\)
\(t=1:\ N=100\times1.5^{1}=150\)
\(t=2:\ N=100\times1.5^{2}=225\)
\(t=4:\ N=100\times1.5^{4}=506.25\approx506\)
\(t=8:\ N=100\times1.5^{8}=2562.89\ldots\approx2563\)
\(\text{Using CASIO calculator with constant multiplier}\)
\begin{array} {|c|c|c|c|}
\hline t & \text{Input} & \text{Output}\ (N) & \text{Rounded}\ (N) \\
\hline {0} & 100= & 100 & 100 \\
\hline {1} & \text{Ans}\times 1.5= & 150 & 150 \\
\hline {2} & = & 225 & 225 \\
\hline {4} & = & 506.25 & 506 \\
\hline {8} & = & 2562.890625 & {2563} \\
\hline \end{array}
b.