IB Mathematics Analysis and Approaches (AA) HL Internal Assessment Exemplar: How can different hull shapes be optimized to minimize material usage while ensuring sufficient buoyancy for a fishing boat?

The student establishes a clear personal rationale for the investigation by linking the topic to childhood observations of boats. This is effective because it gives the exploration a focused purpose and helps the reader understand why these hull shapes matter to the student.

The standardized parameters are a strong organisational choice, but the investigation would be even sharper if the student briefly explained why these specific values were selected as realistic for a fishing boat. That would make the assumptions feel purposeful rather than arbitrary.

The repeated general commentary on round hulls could be shortened. A more concise transition into the parabola model would keep the focus on the mathematics and improve the overall relevance of the exploration.

The statement that the graph shows “3 values” for thickness is not justified by the displayed working. The student should distinguish between graph intersections, roots, and a physically meaningful thickness domain, because claiming three valid thicknesses weakens the logical development.

The conclusion is well signposted and brings the investigation to a clear close. The student summarises the comparative outcome before moving into limitations, which gives the exploration a coherent endpoint.

The use of $${\rho }_{water}\times V_{displaced}=m_{boat}+m_{cargo}$$ is clear and economical. Stating the relation in equation form gives the later calculations a firm mathematical foundation.

The variable table is effective mathematical communication: symbols, meanings, and units are listed together, which helps the reader interpret later equations consistently and reduces ambiguity.

The notation $$V_{inner}=(7+2T)(2.5+2T)(0.7+T)$$ is confusing because the dimensions being enlarged suggest an outer volume, not an inner one. The student should rename and reinterpret this quantity so the algebra matches the geometry being modelled.

The exploration would show stronger personal initiative if the student briefly justified why a fishing boat was the right real-world case and what constraints mattered most. That would make the modelling decisions feel more self-directed.

Choosing to model the round hull in two different ways, first with a parabola and then with a semi-ellipse, shows initiative. This comparison allows the student to test how sensitive the result is to the mathematical representation of the same physical shape.

The student shows independence by comparing multiple hull geometries rather than studying only one design. This indicates purposeful modelling choice and a willingness to test how shape affects material efficiency.

To push the personal approach further, the student could test alternate parameter sets or add a second design criterion such as stability or drag. That would demonstrate more creative exploration beyond repeating the same thickness method across shapes.

Suggesting validation with real data or prototype testing shows initiative beyond the immediate model. The student is thinking about how the mathematics could connect to practical verification, which is a sign of engaged inquiry.

The use of $${\rho }_{water}\times g\times V_{displaced}=m\times g$$ and its simplified form shows appropriate application of Archimedes’ principle. The mathematical setup is relevant and correctly linked to the physical situation.

The expression $$V_b=4T^3+21.8T^2+30.8T$$ is derived from the earlier volume model, but the physical meaning of this quantity is not clearly justified. The student should explain why this equals the boat material volume before using it in the density equation.

The graph caption identifies the cubic function, but the modelling argument would be stronger if the student also stated the domain of thickness being considered. Without a domain, the mathematical result is less clearly connected to the physical problem.

Setting up the definite integral for the parabolic hull is a relevant and appropriate use of calculus. The student correctly translates the geometry into an area calculation rather than relying on a purely descriptive argument.

The step from the shifted curve to $${\int }_0^{2.5}Tdx$$ is mathematically neat but physically simplistic. The student should acknowledge that treating the hull thickness as a constant vertical offset is an approximation, especially for a curved cross-section.

The semi-ellipse model appears to mix geometric parameters and notation in a way that is not fully precise. The student should define the centre and axes with consistent symbols, then show clearly how the equation corresponds to the actual hull dimensions.

A short derivation showing how the semi-ellipse area formula leads to the quadratic would strengthen the mathematics. This would make the work more rigorous and help justify the use of the quadratic formula at HL level.

The symmetry-based integration in the deep V-hull section is mathematically efficient and correctly applied. This is a good example of using structure to simplify the calculation without losing validity.