Applications in 3D (Ex 6E)

Year 10 Mathematics Core · Cambridge Ch 6, §6E · Mr Wong

Name

Class

Date

Learning intentions.

Find a right-angled triangle hidden inside a 3D object (cube, box, pyramid, mast)
Redraw that triangle in 2D, sometimes using Pythagoras first to find a side
Apply SOHCAHTOA to find an unknown length or angle, then report the answer in the 3D context

Key results — handy 3D Pythagoras shortcuts

Cube side $a$: face diag $=a\sqrt{2}$, space diag $=a\sqrt{3}$

Box $a\times b\times c$: base diag $=\sqrt{a^2+b^2}$, space diag $=\sqrt{a^2+b^2+c^2}$

Warm-up

1. A cube has side length $2$ m. With $A$ = front-bottom-left, $B$ = front-bottom-right, $C$ = back-bottom-right and $D$ = back-top-right:

a Exact face diagonal $AC=$

b Exact space diagonal $AD=$

c $\angle DAC=$ (1 d.p.)

d $\angle CAB=$

Part A — Mast and cables

2. A vertical mast is supported at the top by two cables to points $A$ and $B$ on opposite sides. The cable from $A$ is $36$ m at $48°$ to the horizontal; point $B$ is $24$ m from the base. (a) Find the height of the mast (3 d.p.). (b) Find the angle the $B$-cable makes with the horizontal (2 d.p.).

3. A $8$ m vertical pole is supported by four equal cables, each anchored $6$ m from the base of the pole (on the ground). Find (a) the length of one cable, and (b) the angle each cable makes with the ground (2 d.p.).

Part B — Boxes and prisms

4. A rectangular box has dimensions $3 \times 4 \times 5$ cm. Find:

a base diag (3 × 4) =

b space diag =

c angle between space diag and base = (2 d.p.)

5. A rectangular box has dimensions $2 \times 3 \times 6$ m. Find (a) the length of the space diagonal (2 d.p.), and (b) the angle this diagonal makes with the $2\times 3$ base (2 d.p.).

Trig 6E — continued

Part C (pyramid & tent) · Challenge

Name

Part C — Pyramids and tents

6. A right square-based pyramid has base side $6$ m and slant height (apex to midpoint of base edge) $5$ m.

a Vertical height of apex = m

b Angle of slant face to base = (2 d.p.)

7. A ridge tent is set up so the apex (ridge pole) is $2.5$ m above the ground; the canvas comes down to pegs that are $1.8$ m horizontally from a point directly below the ridge. Find the angle the canvas makes with the ground (2 d.p.).

8. A right square-based pyramid has base side $10$ cm and lateral (slant) edge $12$ cm (apex to base corner). Find (a) the vertical height of the apex (2 d.p.), and (b) the angle the slant edge makes with the base (2 d.p.).

Challenge

9. An observer stands $40$ m from the base of a building that is $30$ m tall. A flagpole sits on top of the building. The angle of elevation from the observer to the top of the flagpole is $46.40°$. How tall is the flagpole alone? (2 d.p.) Hint: find the total height, then subtract.

10. A ladder rests with its foot in the corner of a room, leaning against the opposite wall corner. The foot is at the corner; the foot of the wall corner is $1.2$ m along one wall and $1.5$ m along the other wall, and the top of the ladder is $3$ m up. Find the length of the ladder (2 d.p.).