Using Trigonometry in 3D Problems
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Using Trigonometry in 3D Problems — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionCombining Pythagoras and Trig in 3D GeometryAssessment Practice
8 marks~12 minCriterion D
A designer models a square-based pyramid tent. The base is a square of side 6 m. Each triangular face has a slant height (the perpendicular distance from the midpoint of a base edge to the apex, measured along the face) of 8 m.

The diagram shows the tent with its key measurements labelled.
a
Show that the vertical height of the tent is h=55h = \sqrt{55} m. [2]
b
Calculate the total canvas area of the four triangular faces. Give your answer in m2^2 to 3 significant figures.

The designer claims the total canvas area needed is at least 95 m2^2. Justify whether the designer's claim is correct. [3]
c
In reality, the measured vertical height of a prototype tent is 7.2 m. Use the formula below to calculate the percentage error between the model's predicted height and the measured height. Give your answer to 3 significant figures.

percentage error=∣predicted−actual∣actual×100%\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{\text{actual}} \times 100\%

Advise the designer whether the model is sufficiently reliable to use in production, justifying your advice with reference to your calculated percentage error. [3]
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Solutions

2QuestionApplications in Architecture and PhysicsAssessment Practice
8 marks~12 minCriterion C
The diagram shows a vertical tower of height hh metres standing on horizontal ground. An observer at point P measures the angle of elevation θ\theta to the top of the tower from four different horizontal distances dd metres from the base of the tower.

The recorded data are shown below.

θ\theta (degrees): 15, 30, 45, 60

dd (metres): 37.32, 17.32, 10.00, 5.77
a
Show that d=htan⁡θd = \dfrac{h}{\tan\theta}, by referring to the right-angled triangle formed by the tower, the ground, and the line of sight. [2]
b
Using the row θ=45°\theta = 45° from the table, find the height hh of the tower. Hence find the horizontal distance dd when θ=20°\theta = 20°. Give both answers in metres to 3 significant figures. [3]
c
A student claims: "As the angle of elevation doubles, the horizontal distance halves."

Using at least two pairs of values from the table, justify whether this claim should be accepted or rejected. Your justification must refer to the formula d=htan⁡θd = \dfrac{h}{\tan\theta} to explain why the relationship between dd and θ\theta is or is not proportional in the way the student describes. [3]
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Solutions

3QuestionCalculating Diagonals and Angles in CuboidsAssessment Practice
10 marks~15 minCriterion A
A rectangular box has a base measuring 33 cm by 44 cm and a height of hh cm. The space diagonal of the box has length 1313 cm.

The diagram shows the box with vertices AA, BB, CC, DD on the base and A′A', B′B', C′C', D′D' directly above them, where ACAC is the base diagonal and AC′AC' is the space diagonal.
a
Show that h=12h = 12 cm. [3]
b
Find the angle between the space diagonal AC′AC' and the base diagonal ACAC. Give your answer correct to one decimal place. [3]
c
A second rectangular box is geometrically similar to the first, with a space diagonal of length 2626 cm. The manufacturer claims the second box has a total surface area of 384384 cm2^2. Justify whether this claim is correct. [4]
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Solutions

4QuestionCalculating Diagonals and Angles in CuboidsAssessment Practice
10 marks~15 minCriterion B
The table below shows three cuboids with integer edge lengths.

Base length aa (cm)358
Base width bb (cm)41215
Height hh (cm)51317


For each cuboid, the space diagonal connects one corner of the base to the diagonally opposite corner of the top face. The angle θ\theta is measured between the space diagonal and the base plane.
a
Find the base diagonal dd and the angle θ\theta for each cuboid. Give your answers for θ\theta to the nearest degree. [3]
b
Show that tan⁡θ=1\tan\theta = 1 for every cuboid in the table, and hence write a general formula for θ\theta in terms of aa, bb, and hh. [3]
c
A fourth cuboid has base length a=6a = 6 cm, base width b=6b = 6 cm, and space diagonal of length 10 cm. Justify whether θ\theta for this cuboid is greater than, equal to, or less than 45∘45^\circ, without using a calculator. [4]
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