In each of the problems below, show the different stages of mathematical modelling for solving the problems. 2. Suppose the adjoining figure represents an aerial photograph of a forest with each dot representing a tree. Your purpose is to find the number of trees there are on this tract of land as part of an environmental census.

Understand the Question Counting every tree individually on an aerial photograph is impractical. We use mathematical modelling via quadrat sampling: 1. Divide the total area into a uniform grid of equal-sized smaller squares. 2. Count the trees in a few randomly selected sample squares. 3. Calculate the mean (average) number of trees per sample square. 4. Multiply the average by the total number of squares to estimate the total tree count. Step 1 · Formulate the Model (Grid Division) Divide the entire photograph into a uniform grid of equal squares. Total number of squares, Number of randomly selected sample squares, Step 2 · Data Collection (Sample Counts) Count the number of trees in the randomly chosen squares. Let the sample counts be: Step 3 · Calculate the Total Estimated Trees Find the total trees in the sample squares Find the average number of trees per square Estimate the total tree population across all squares Step 4 · Interpretation and Refinement The calculated value of trees is an approximation. The model assumes a relatively uniform distribution of trees across the tract. To increase accuracy and reduce error, we can: Increase the number of sample squares (larger sample size ). Use stratified sampling if certain regions (e.g. riverbanks or dense forest patches) have significantly different tree densities. Answer > Estimated number of trees Common Mistakes Counting all dots: Trying to count every single dot is impractical; the core objective of mathematical modelling here is sampling and statistical estimation. Non-random sampling: Selecting only dense or sparse squares leads to overestimation or underestimation; sample squares must be randomly selected across the grid. Ignoring distribution variations: Assuming the estimate is exact rather than an approximation that depends on sample size and tree uniformity.

Appendix 2: Mathematical Modelling | A2.2

Question 2

In each of the problems below, show the different stages of mathematical modelling for solving the problems.

Suppose the adjoining figure represents an aerial photograph of a forest with each dot representing a tree. Your purpose is to find the number of trees there are on this tract of land as part of an environmental census.

Solution

Understand the Question

Counting every tree individually on an aerial photograph is impractical.
We use mathematical modelling via quadrat sampling:
1. Divide the total area into a uniform grid of equal-sized smaller squares.
2. Count the trees in a few randomly selected sample squares.
3. Calculate the mean (average) number of trees per sample square.
4. Multiply the average by the total number of squares to estimate the total tree count.

Step 1 · Formulate the Model (Grid Division)

Divide the entire photograph into a uniform grid of $10 \times 10 = 100$ equal squares. Diagram 1

Total number of squares, $N = 100$
Number of randomly selected sample squares, $n = 5$

Step 2 · Data Collection (Sample Counts)

Count the number of trees in the $5$ randomly chosen squares. Let the sample counts be: $25, 30, 28, 27, 32$

Step 3 · Calculate the Total Estimated Trees

Find the total trees in the sample squares

\begin{aligned} \text{Total trees in sample} &= 25 + 30 + 28 + 27 + 32 \\ &= 142 \end{aligned}

Find the average number of trees per square

\begin{aligned} \text{Average trees per square} &= \dfrac{\text{Total trees in sample}}{\text{Number of sample squares}} \\[0.6em] &= \dfrac{142}{5} \\[0.6em] &= 28.4 \end{aligned}

Estimate the total tree population across all $100$ squares

\begin{aligned} \text{Estimated total trees} &= \text{Average trees per square} \times \text{Total number of squares} \\[0.6em] &= 28.4 \times 100 \\[0.6em] &= 2840 \end{aligned}

Step 4 · Interpretation and Refinement

The calculated value of $2840$ trees is an approximation. The model assumes a relatively uniform distribution of trees across the tract.

To increase accuracy and reduce error, we can:

Increase the number of sample squares (larger sample size $n$ ).
Use stratified sampling if certain regions (e.g. riverbanks or dense forest patches) have significantly different tree densities.

Answer

Estimated number of trees $\approx 2840$

Common Mistakes

Counting all dots: Trying to count every single dot is impractical; the core objective of mathematical modelling here is sampling and statistical estimation.
Non-random sampling: Selecting only dense or sparse squares leads to overestimation or underestimation; sample squares must be randomly selected across the grid.
Ignoring distribution variations: Assuming the estimate is exact rather than an approximation that depends on sample size and tree uniformity.

Question 2

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