Higher June 2024 Paper 2 Q25
25 Here is a right-angled triangular prism.
The ratio of the edges is \(\quad a : b : c : d = 3 : 4 : 5 : 12\)
The volume of the prism is \(1125\ \text{cm}^3\)
Work out the total length of all of the edges of the prism. [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: works out a scale factor | ||
| \(\dfrac{1}{2} \times 3(L) \times 4(L) \times 12(L)\) or \(72(L^3)\) where \(L\) is any variable or any positive value | M1 | oe volume eg \((L = 2)\ \dfrac{1}{2} \times 6 \times 8 \times 24\) or 576 |
| \(1125 \div\) their 72 or \(\dfrac{125}{8}\) or 15.625 | M1dep | oe eg \(1125 \times 2 \div (3 \times 4 \times 12)\) eg \((L = 2)\ 1125 \div\) their 576 or \(\dfrac{125}{64}\) |
| \(\sqrt[3]{\text{their } \dfrac{125}{8}}\) or \(\dfrac{5}{2}\) or 2.5 | M1dep | oe eg \((L = 2)\ \sqrt[3]{\text{their } \dfrac{125}{64}}\) or \(\dfrac{5}{4}\) or 1.25 |
| \(2 \times 3 \times\) their \(2.5 + 2 \times 4 \times\) their 2.5 \(+\ 2 \times 5 \times\) their 2.5 \(+\ 3 \times 12 \times\) their 2.5 | M1dep | oe eg \((L = 2)\) \(2 \times 6 \times\) their \(1.25 + 2 \times 8 \times\) their 1.25 \(+\ 2 \times 10 \times\) their 1.25 \(+\ 3 \times 24 \times\) their 1.25 |
| 150 | A1 | SC4 [119, 119.1] |
| Alternative method 2: works out a value of \(a\), \(b\), \(c\) or \(d\) | ||
| Correct expression for volume in terms of \(a\) or \(b\) eg \(\dfrac{1}{2} \times a \times \dfrac{4a}{3} \times \dfrac{12a}{3}\) or \(\dfrac{8a^3}{3}\) or \(\dfrac{1}{2} \times \dfrac{3b}{4} \times b \times \dfrac{12b}{4}\) or \(\dfrac{9b^3}{8}\) | M1 | oe in terms of \(c\) or \(d\) eg \(\dfrac{1}{2} \times \dfrac{3c}{5} \times \dfrac{4c}{5} \times \dfrac{12c}{5}\) or \(\dfrac{72c^3}{125}\) or \(\dfrac{1}{2} \times \dfrac{3d}{12} \times \dfrac{4d}{12} \times d\) or \(\dfrac{d^3}{24}\) may be implied by an equation eg \(a \times \dfrac{4a}{3} \times \dfrac{12a}{3} = 1125 \times 2\) |
| \(a^3 = 1125 \div\) their \(\dfrac{8}{3}\) or \(a^3 = \dfrac{3375}{8}\) or \(b^3 = 1125 \div\) their \(\dfrac{9}{8}\) or \(b^3 = 1000\) | M1dep | oe eg \(c^3 = 1125 \div\) their \(\dfrac{72}{125}\) or \(c^3 = \dfrac{15\,625}{8}\) or \(d^3 = 1125 \div\) their \(\dfrac{1}{24}\) or \(d^3 = 27\,000\) |
| \(a = \sqrt[3]{\text{their } \dfrac{3375}{8}}\) or \(a = 7.5\) or \(b = \sqrt[3]{1000}\) or \(b = 10\) | M1dep | oe eg \(c = \sqrt[3]{\text{their } \dfrac{15\,625}{8}}\) or \(c = 12.5\) or \(d = \sqrt[3]{27\,000}\) or \(d = 30\) |
| \(2 \times\) their \(a + 2 \times \dfrac{4}{3} \times\) their \(a\) \(+\ 2 \times \dfrac{5}{3} \times\) their \(a + 3 \times \dfrac{12}{3} \times\) their \(a\) or \(2 \times \dfrac{3}{4} \times\) their \(b + 2 \times\) their \(b\) \(+\ 2 \times \dfrac{5}{4} \times\) their \(b + 3 \times 3 \times\) their \(b\) | M1dep | oe correct method using their \(c\) or their \(d\) |
| 150 | A1 | SC4 [119, 119.1] |
Additional guidance
Up to M3 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts