Using Trigonometry in 3D Problems
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Using Trigonometry in 3D Problems — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionApplications in Architecture and PhysicsAssessment Practice
6 marks~9 minCriterion A
The diagram shows a right-angled triangle ABCABC representing a roof truss, where ∠B=90°\angle B = 90°. The horizontal span AB=8AB = 8 m, the vertical rise BC=3BC = 3 m, and the hypotenuse AC=hAC = h m.
a
Show that h=73h = \sqrt{73}. [1]
b
A second roof truss is geometrically similar to the first. Its hypotenuse is 2732\sqrt{73} m. Find the horizontal span of the second truss, in metres. [2]
c
A third roof truss has the same horizontal span of 88 m but a vertical rise rr m, where r>3r > 3, and a hypotenuse of 89\sqrt{89} m. Find the value of rr and hence justify whether this third truss meets a building regulation requiring a vertical rise of at least 55 m. [3]
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2QuestionApplications in Architecture and PhysicsAssessment Practice
6 marks~9 minCriterion C
A vertical pole of height 10 m stands on horizontal ground. A sun ray makes an elevation angle θ\theta with the horizontal, casting a shadow of length LL metres along the ground.

The ratio RR is defined as
R=10L.R = \frac{10}{L}.
a
Write down the value of RR when θ=45°\theta = 45°. [1]
b
Calculate RR to 3 significant figures for each elevation angle in the table below.

Elevation angle (degrees): 30, 40, 50, 60

Shadow length LL (m): 17.32, 11.92, 8.39, 5.77

R=10LR = \dfrac{10}{L}: ?, ?, ?, ? [2]
c
A student claims that R=tan⁡(θ)R = \tan(\theta) for any elevation angle θ\theta. Justify this claim using your completed table, and hence deduce whether the model R=tan⁡(θ)R = \tan(\theta) is reliable enough to predict shadow behaviour at θ=70°\theta = 70° for a solar panel alignment system that requires predictions accurate to 3 significant figures. [3]
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3QuestionApplications in Architecture and PhysicsAssessment Practice
9 marks~14 minCriterion B
The table shows the dimensions and space diagonal lengths for a sequence of rectangular prisms.

Prism number1234
Dimensions (cm)1×1×11\times1\times11×2×31\times2\times31×3×51\times3\times51×4×71\times4\times7
Space diagonal (cm)3\sqrt{3}14\sqrt{14}35\sqrt{35}d4d_4


The space diagonal of a rectangular prism with dimensions ll, ww, hh is given by
d=l2+w2+h2.d = \sqrt{l^2 + w^2 + h^2}.
a
Calculate the exact value of d4d_4, the space diagonal of Prism 4. Give your answer in the form k\sqrt{k}, where kk is an integer. [2]
b
Deduce a rule for dnd_n, the space diagonal of Prism nn, in the form
dn=an2+bn+cd_n = \sqrt{an^2 + bn + c}
where aa, bb, and cc are integers. State the values of aa, bb, and cc. [3]
c
Verify your rule for Prism 3. Hence justify why the expression under the square root must be a quadratic function of nn for every prism in this sequence, referring to the structure of the dimensions. [4]
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4QuestionApplications in Architecture and PhysicsAssessment Practice
9 marks~14 minCriterion D
An architect is designing a roof truss for a community centre. The truss forms an isosceles triangle in a vertical plane with a horizontal base of 12 m. The pitch angle (the angle between the base and each sloping rafter) is 35∘35^\circ. The ridge height above the base is hh m and each rafter has length rr m.
a
Show that h=6tan⁡35∘h = 6\tan 35^\circ, and hence calculate hh to 3 significant figures. Give your answer in metres. [2]
b
Find the rafter length rr to 3 significant figures. Give your answer in metres. [2]

The architect uses the linear model H(x)=xtan⁡35∘H(x) = x\tan 35^\circ to predict the height (in metres) at horizontal distance xx metres from the left support. After construction, three on-site measurements are taken.

xx (m): 3, 6, 9

Predicted H(x)H(x) (m): 2.10, 4.20, 6.30

Measured height (m): 2.61, 4.88, 7.43
c
Calculate the percentage error at x=6x = 6 m, using

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

Give your answer to 1 decimal place. [2]
d
Assess whether the linear model H(x)=xtan⁡35∘H(x) = x\tan 35^\circ is suitable for use in the final construction drawings, referring to the percentage errors at all three values of xx and giving one reason why the measured heights may differ from the model predictions. [3]
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