IB Maths AI SL 3.1 Three-dimensional geometry Question Bank

Practise IB Maths AI SL 3.1 by calculating distances, areas and volumes from three-dimensional coordinates and diagrams.

Exam points

Use coordinates or a diagram to calculate a distance, area or volume in three dimensions.
Interpret a three-dimensional measurement in context and include the correct units.
Check a geometric result using a second dimension, scale or physical constraint.

[Maximum number: 5]

A hollow chocolate box is manufactured in the form of a right prism with a regular hexagonal base. The height of the prism is $h \mathrm{~cm}$ , and the top and base of the prism have sides of length $x \mathrm{~cm}$ .

Figure for Question SL 3.1—Three-dimensional geometry question 1 — IB Maths AI SL

Given that the total external surface area of the box is $1200 \mathrm{~cm}^{2}$ , show that the volume of the box may be expressed as $V=300 \sqrt{3} x-\frac{9}{4} x^{3}$ .

total surface area of prism $1200=2\left(3 x^{2} \frac{\sqrt{3}}{2}\right)+6 x h$ M1A1

Note: Award M1 for expressing total surface areas as a sum of areas of rectangles and hexagons, and A1 for a correctly substituted formula, equated to 1200.

\begin{aligned} & h=\frac{400-\sqrt{3} x^{2}}{2 x} \\ & \text { volume of prism }=\frac{3 \sqrt{3}}{2} x^{2} \times h \\ & =\frac{3 \sqrt{3}}{2} x^{2}\left(\frac{400-\sqrt{3} x^{2}}{2 x}\right) \\ & =300 \sqrt{3} x-\frac{9}{4} x^{3} \end{aligned}

Note:The AG line must be seen for the final A1 to be awarded.

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SL 3.1—Three-dimensional geometry question 1