15.4 The base of a pyramid is a triangle with sides $3 \mathrm{~cm}, p \mathrm{~cm}$ and $5 \mathrm{~cm}$. If the height and volume of the pyramid are $q \mathrm{~cm}$ and $12 \mathrm{~cm}^{3}$ respectively, find $q$.
$3^{2}+4^{2}=5^{2} \Rightarrow$ the triangle is a right-angled triangle (Converse, Pythagoras' theorem)
Base area $=\frac{1}{2} \cdot 3 \cdot 4 \mathrm{~cm}^{2}=6 \mathrm{~cm}^{2}$
Volume $=\frac{1}{3} \cdot 6 \cdot q \mathrm{~cm}^{3}=12 \mathrm{~cm}^{3} \Rightarrow q=6$