IB Mathematics Analysis and Approaches (AA) HL Internal Assessment Exemplar: Modeling and Calculating the Surface Area of a Plastic Container

The table of contents lacks page references, which hinders navigation. Consider adding consistent page numbers for each section.

Citing Dawkins without a full reference makes sourcing unclear. Include a properly formatted bibliography entry here.

The environmental commentary, while interesting, digresses from the mathematical aim. Consider condensing to focus on your calculation objectives.

Figure 6’s segmentation boundaries are not explicitly annotated in the image. Add labels for sections A–H directly on the diagram for clarity.

Referring to the container as a “ceramic pot” in Section B seems to be a copy–paste error. Correct terminology to maintain coherence.

The calculation for Section F on page 14 is truncated and unreadable. Ensure full equations are displayed and proofs are complete.

Bibliography entries show inconsistent formatting. Adopt a single citation style (e.g., APA) for uniform presentation.

Figures 2 and 3 lack axis labels and scales, which reduces clarity of the approximations. Label axes to improve multiple representation effectiveness.

The expression $$\Delta s=\sqrt{(x_1-x_0{)}^2+(y_1-y_0)}$$ omits the square on the y-difference term. It should read $$\sqrt{(x_1-x_0{)}^2+(y_1-y_0{)}^2}$$.

The Lagrange interpolation formula is presented correctly and labels are clear, showing effective use of mathematical notation.

In Section A, the domain is miswritten as $$0\le y\le 2.2$$ instead of $$0\le x\le 2.2$$, which may confuse readers.

Table 3 uses symbols χ and ν which are unexplained and nonstandard here. Use x and y headings or clarify these symbols.

In the surface area calculation (page 11), the derivative notation $$\frac{dy}{dx}$$ is followed by ‘dx’, which is incorrect. Remove the extra ‘dx’.

Several intermediate sums lack units (e.g., 54.07), which may confuse the cumulative result. Include cm² in each result.

The introduction effectively connects personal motivation to environmental context, showing clear engagement with the problem.

Using Geogebra to scale and plot points demonstrates initiative and significantly reduces manual error.

Excluding the base and top from the surface calculation needs justification. How will this affect the total area you report?

The weaknesses section acknowledges human error but lacks quantitative impact analysis. Consider estimating the error margin in cm².

The conclusion restates results but could reflect on precision by comparing with direct measurement of the container’s label.

The derivation of the surface area integral formula is precise and correctly uses $${\int }_a^{b}2\pi y\sqrt{1+(\frac{dy}{dx}{)}^2}dx$$.

The substitution into the surface area formula incorrectly writes $$\sqrt{1+(f^{\prime }(x))}$$ without squaring. The integrand should include $$\sqrt{1+(f^{\prime }(x){)}^2}$$.

The step-by-step algebraic manipulation to the limit process is thorough and shows strong understanding of arc-length integration.

Assuming all container sections follow polynomial profiles should be justified. Consider validating the choice with residual analysis or fit error.

Relying on an online calculator for polynomial fits without showing any manual verification reduces the demonstration of understanding.

The calculus integral steps to compute each section’s area are methodical and show effective use of relevant mathematics.