Problem-solving Pack

Mathematics

Year 9 · Architecture

Problem-solving Pack

Name: _________________________________

Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 36

Leaning ladder. A ladder of length 10 m leans against a vertical wall. The foot of the ladder is 3 m from the base.

(a) Sketch the situation.
(b) Find the height reached, to 2 d.p.
(c) If the wall is 12 m tall, find the distance from the top of the ladder to the top of the wall.

Working space

2Problem 2 of 36

Pythagorean triples. Pythagorean triples are sets of three positive integers

(a, b, c)

satisfying

a^2 + b^2 = c^2

.

(a) Verify

(3, 4, 5)

(5, 12, 13)

(8, 15, 17)

are triples.
(b) Show

(7, 24, 25)

is also a triple.
(c) Multiply

(3, 4, 5)

by 7 to find another triple.
(d) Show that if

(a, b, c)

is a triple, then so is

(ka, kb, kc)

for any positive integer

k

Working space

3Problem 3 of 36

3D Pythagoras. A cuboid has length 4, width 3, height 2 m.

(a) Find the diagonal of the base.
(b) Find the space diagonal (corner to opposite corner).
(c) Find the angle between the space diagonal and the base.

Working space

4Problem 4 of 36

Surds simplification. Simplify each:

(a)

\sqrt{72}

(b)

\sqrt{12} + \sqrt{27}

(c)

\sqrt{50} \times \sqrt{2}

(d)

\dfrac{8}{\sqrt{2}}

(rationalised)

Working space

5Problem 5 of 36

Diagonal of a cube. A cube has side length

a

.

(a) Find the face diagonal in terms of

a

.
(b) Find the space diagonal.
(c) For

a = 5

cm, compute both diagonals.

Working space

6Problem 6 of 36

The fish tank. A fish tank is 60 × 40 × 30 cm. A diagonal frame is built from corner to opposite corner.

(a) Find the length of the frame.
(b) Find the angle of the frame above the bottom.
(c) Find the lengths needed for two diagonals across each face.

Working space

7Problem 7 of 36

Triangle perimeter with surds. A triangle has vertices

A(0, 0), B(3, 4), C(0, 4)

.

(a) Find each side.
(b) Show one side is a Pythagorean triple-like length.
(c) Find the perimeter.

Working space

8Problem 8 of 36

Diagonal of a rectangular field. A rectangular field 60 m × 80 m has a diagonal path.

(a) Find the path length.
(b) Find the angle the path makes with the longer side.
(c) The field is enlarged by a scale factor of 1.5. Find the new diagonal length.

Working space

9Problem 9 of 36

Surds in algebra. A square has area

50

cm².

(a) Find the side length as a simplified surd.
(b) Find the diagonal of the square.
(c) Find the perimeter.

Working space

10Problem 10 of 36

Application: angle of elevation. A pole of height 12 m casts a shadow 5 m long.

(a) Find the distance from the tip of the shadow to the top of the pole.
(b) Find the angle of elevation of the sun.
(c) Identify any Pythagorean triple.

Working space

11Problem 11 of 36

Rationalising denominators. Rationalise each:

(a)

\dfrac{5}{\sqrt{2}}

(b)

\dfrac{1}{\sqrt{3} - 1}

(c)

\dfrac{3}{\sqrt{5} + \sqrt{2}}

Working space

12Problem 12 of 36

Pythagoras in space. A box has dimensions

a, b, c

. The space diagonal

d

satisfies

d^2 = a^2 + b^2 + c^2

.

(a) Express

d

in surd form for

a = 1, b = 2, c = 2

.
(b) Find the smallest cube containing a stick of length 10 m.
(c) Investigate: can the space diagonal of a cube equal twice the side length?

Working space

13Problem 13 of 36

Cone and cylinder. A cone has radius 6 cm and height 8 cm. A cylinder has the same radius and is made of the same volume of material.

(a) Find the volume of the cone (in terms of π).
(b) Find the height of the cylinder.

Working space

14Problem 14 of 36

Pyramid at Giza. A square-based pyramid has base side 230 m and vertical height 147 m.

(a) Find the volume.
(b) Find the length of one slanted edge from a base vertex to the apex.

Working space

15Problem 15 of 36

Surface area of a sphere. A glass sphere has external SA ≈ 31 416 m².

(a) Find the external radius.
(b) If the glass is 15 cm thick, find the internal volume (3 s.f.).

Working space

16Problem 16 of 36

Fish tank. A tank 60 × 40 × 30 cm is 33% full. Water is poured into a cylindrical container with radius 20 cm. Find the depth of water in the cylinder.

Working space

17Problem 17 of 36

Rearranging formulas.

V = (4/3)\pi r^3

.

(a) Make

r

the subject.
(b) Find

r

for

V = 36\pi

Working space

18Problem 18 of 36

Pizza box from a square sheet. Six 4 cm × 4 cm squares are cut from corners of a square sheet, which is then folded into a square prism (open-top). The base is

x

cm ×

x

cm.

(a) Height of box = ?
(b) Volume = 1600 cm³. Find

x

Working space

19Problem 19 of 36

Volume of a frustum. A cone has radius 9 and height 12 cm. The top is sliced parallel to the base at a height of 8 cm, leaving a frustum.

(a) Find the radius of the new top.
(b) Find the volume of the frustum.

Working space

20Problem 20 of 36

Cylinder + hemisphere. A solid is a cylinder with a hemisphere on top, both of radius 5 cm. Cylinder height = 10 cm.

(a) Find the total volume.
(b) Find the total surface area (don't count the joined circle).

Working space

21Problem 21 of 36

Trigonometry on a square pyramid. PQRS is the square base of a pyramid with apex V. Square sides 8 cm; vertical height VG = 12 cm; M is the midpoint of QR.

(a) Write down GM.
(b) Find VM.
(c) Find the angle between face VQR and the base.

Working space

22Problem 22 of 36

Capacity puzzle. A cone-shaped funnel has top radius 4 cm and height 10 cm. Water pours in from a tap at 50 cm³/s.

(a) Find the funnel's capacity.
(b) Find the time to fill the funnel.
(c) The funnel pours into a cylindrical glass (r = 5, h = 8). Will the glass overflow when the funnel fully empties into it?

Working space

23Problem 23 of 36

Composite shape volume. A solid is composed of a hemisphere of radius 6 cm on top of a cube of side 12 cm.

(a) Find the total volume.
(b) Find the total exposed surface area.

Working space

24Problem 24 of 36

Volume scaling. Two similar pyramids have a linear scale factor of 2.

(a) Find the volume scale factor.
(b) The smaller has volume 50 cm³. Find the larger volume.
(c) The larger has surface area 144 cm². Find the smaller surface area.

Working space

25Problem 25 of 36

Tree height. A tree casts a 5 m shadow when the angle of elevation of the sun is 40°.

(a) Find the tree height.
(b) Find the slant distance from the tip of the shadow to the top of the tree.

Working space

26Problem 26 of 36

Building observation. A pedestrian 1.7 m tall stands 40 m from a 31 m monument.

(a) Find the angle of elevation of the top of the monument from the pedestrian's eyes.
(b) The pedestrian moves to 20 m from the monument. Find the new angle.

Working space

27Problem 27 of 36

Zipline. A zipline runs from a 154 m UN building to a 443 m Empire State Building, separated horizontally by 1488 m.

(a) Find the zipline length.
(b) Find the angle of elevation from the UN to the ESB.

Working space

28Problem 28 of 36

Bearings. A boat sails 12 km on a bearing of 60°.

(a) Find the easterly displacement.
(b) Find the northerly displacement.
(c) The boat then sails 10 km on bearing 150°. Find the resulting position.

Working space

29Problem 29 of 36

Pyramid trigonometry. A pyramid has square base side 10 cm, apex 12 cm above the centre.

(a) Find the half-diagonal of the base.
(b) Find the slant edge length.
(c) Find the angle the slant edge makes with the base.

Working space

30Problem 30 of 36

Cuboid diagonal trigonometry. A cuboid has length 8, width 6, height 4.

(a) Find the base diagonal.
(b) Find the space diagonal.
(c) Find the angle the space diagonal makes with the base.

Working space

31Problem 31 of 36

Exact-value triangles. For a 30-60-90 triangle with shorter leg 1:

(a) Find the hypotenuse.
(b) Find the longer leg.
(c) State the ratios sin, cos, tan for 30° and 60°.

Working space

32Problem 32 of 36

45-45-90 triangle. A square has side 1.

(a) Find the diagonal.
(b) Find sin 45°, cos 45°, tan 45°.

Working space

33Problem 33 of 36

Angle of depression problem. A coastguard 50 m above sea level sees a boat with angle of depression 8°.

(a) Find the horizontal distance to the boat.
(b) The boat moves so the angle is 12°. Find the new distance.
(c) How fast does the boat's distance to the coastguard's vertical change?

Working space

34Problem 34 of 36

Find the side or angle. A right triangle has hyp 20 cm and angle 25°.

(a) Find both legs.
(b) Find the area.
(c) Find the perimeter.

Working space

35Problem 35 of 36

3D trig in a cuboid. A box is 12 cm × 8 cm × 6 cm. A diagonal is drawn from one corner to the opposite corner.

(a) Find the base diagonal.
(b) Find the space diagonal.
(c) Find the angle between the space diagonal and the longest face.

Working space

36Problem 36 of 36

Application: roof slope. A house has a roof that rises 3 m over a horizontal span of 5 m.

(a) Find the angle of the roof above horizontal.
(b) Find the slant length of the roof.
(c) Find the area of one side of the roof, if the house is 10 m long.

Working space