7 Vijay is giving clues to his friend Henry about the dimensions of a cuboid. The length, width and height are all whole numbers in centimetres. Clue 1: Vijay says: 'It has a volume of \( 96 \mathrm{~cm}^{3} \).' Henry says ' think it has a length of 48 cm , a width of 2 cm and a height of 1 cm .' Vijay tells Henry he is wrong. Clue 2: Vijay says: 'The width and height are the same.' Henry says 'I think it has a length of 24 cm , a width of 2 cm and a height of 2 cm .' Vijay tells Henry he is wrong again. Clue 3: Vijay says: 'The length, width and height are all more than 1 cm .' What should Henry's next guess be? He should not need any more clues.

To find the dimensions of the cuboid with a volume of \( 96 \, \text{cm}^3 \), we can denote the length as \( l \), the width as \( w \), and the height as \( h \). According to the clues: 1. The volume is given by the equation: \[ l \times w \times h = 96 \] 2. From Clue 2, we know that the width and height are the same, so we can set \( w = h \). Thus, we can rewrite the volume equation as: \[ l \times w^2 = 96 \] 3. From Clue 3, we know that \( l, w, h > 1 \). Now, we can find the possible values for \( w \) and \( l \) by factoring \( 96 \). The prime factorization of \( 96 \) is: \[ 96 = 2^5 \times 3^1 \] Now, we can find the pairs of factors of \( 96 \) that can be expressed as \( l \times w^2 \). Let's check the possible values for \( w \): - If \( w = 2 \): \[ l \times 2^2 = 96 \implies l \times 4 = 96 \implies l = 24 \] (This was Henry's second guess, which was incorrect.) - If \( w = 3 \): \[ l \times 3^2 = 96 \implies l \times 9 = 96 \implies l = \frac{96}{9} \text{ (not a whole number)} \] - If \( w = 4 \): \[ l \times 4^2 = 96 \implies l \times 16 = 96 \implies l = 6 \] (This gives us \( l = 6 \), \( w = 4 \), \( h = 4 \).) - If \( w = 5 \): \[ l \times 5^2 = 96 \implies l \times 25 = 96 \implies l = \frac{96}{25} \text{ (not a whole number)} \] - If \( w = 6 \): \[ l \times 6^2 = 96 \implies l \times 36 = 96 \implies l = \frac{96}{36} \text{ (not a whole number)} \] - If \( w = 8 \): \[ l \times 8^2 = 96 \implies l \times 64 = 96 \implies l = \frac{96}{64} \text{ (not a whole number)} \] - If \( w = 12 \): \[ l \times 12^2 = 96 \implies l \times 144 = 96 \implies l = \frac{96}{144} \text{ (not a whole number)} \] - If \( w = 16 \): \[ l \times 16^2 = 96 \implies l \times 256 = 96 \implies l = \frac{96}{256} \text{ (not a whole number)} \] The only valid whole number solution we found is: - \( l = 6 \) - \( w = 4 \) - \( h = 4 \) Thus, Henry's next guess should be: \[ \text{Length: } 6 \, \text{cm}, \text{ Width: } 4 \, \text{cm}, \text{ Height: } 4 \, \text{cm} \] So, Henry should guess \( 6 \, \text{cm}, 4 \, \text{cm}, 4 \, \text{cm} \).