Higher November 2018 Paper 3 Q23
23 The diagram shows the side view of a step ladder with a horizontal strut of length 48 cm
The strut is one third of the way up the ladder.
The symmetrical cross section of the ladder shows two similar triangles.
Not drawn accurately
Work out the vertical height, \(h\) cm, of the ladder. [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(48 \div 2 \times 3\) or 72 | M1 | oe |
| \(\text{their } 72 \div 2\) or 36 | M1dep | \(\cos^{-1}\left(\dfrac{36}{141}\right)\) or 75.2 |
| \(141^2 - \text{their } 36^2\) or 18 585 | M1dep | ft their base \(\div\) 2 \(\sin(\text{their } 75.2) = \dfrac{h}{141}\) or \(\tan(\text{their } 75.2) = \dfrac{h}{\text{their } 36}\) |
| \(\sqrt{141^2 - \text{their } 36^2}\) or \(\sqrt{18\,585}\) | M1dep | \(141 \times \sin(\text{their } 75.2)\) or \(\text{their } 36 \times \tan(\text{their } 75.2)\) |
| [136.2, 136.4] or 136 | A1 | |
| Alternative method 2 | ||
| \(141 \div 3\) or 47 | M1 | oe |
| 24 and \(\text{their } 47 \times 2\) or 24 and 94 or 12 and their 47 | M1dep | \(\cos^{-1}\left(\dfrac{24}{94}\right)\) or 75.2 |
| \(\text{their } 94^2 - 24^2\) or 8260 or \(\sqrt{8260}\) or 90.88… or \(\text{their } 47^2 - 12^2\) or 2065 or \(\sqrt{2065}\) or 45.44… | M1dep | \(\sin(\text{their } 75.2) = \dfrac{h}{\text{their } 94}\) or \(\tan(\text{their } 75.2) = \dfrac{h}{24}\) |
| \(\sqrt{\text{their } 94^2 - 24^2} \times 3 \div 2\) or \(\sqrt{8260} \times 3 \div 2\) or \(90.88… \times 3 \div 2\) or \(\sqrt{\text{their } 47^2 - 12^2} \times 3\) or \(\sqrt{2065} \times 3\) or \(45.44… \times 3\) | M1dep | \(\text{their } 94 \times \sin(\text{their } 75.2) \times 3 \div 2\) or \(24 \times \tan(\text{their } 75.2) \times 3 \div 2\) |
| [136.2, 136.35] or 136 | A1 | |
Additional guidance
Values may be seen on diagram in correct positions