직육면체 부피

$\displaystyle {\overline{\mathrm{AB}}} = a$, $\displaystyle {\overline{\mathrm{BF}}} = b \left( a > b \right)$라 하자. 직사각형 ${\mathrm{RSGH}}$에서 $\displaystyle {\overline{\mathrm{SG}}} = a - b$, $\displaystyle {\overline{\mathrm{GH}}} = a$이므로 $\displaystyle {\overline{\mathrm{SH}}} ^{2}$$\displaystyle = {\overline{\mathrm{SG}}} ^{2} + {\overline{\mathrm{GH}}} ^{2}$$= \left( a - b \right) ^{2} + a ^{2}$$= 2 a ^{2} - 2 ab + b ^{2}$ 직각삼각형 ${\mathrm{DHS}}$에서 $\displaystyle {\overline{\mathrm{SD}}} ^{2}$$\displaystyle = {\overline{\mathrm{SH}}} ^{2} + {\overline{\mathrm{DH}}} ^{2}$ $= \left( 2 a ^{2} - 2 ab + b ^{2} \right) + b ^{2}$$= 2 \left( a ^{2} - ab + b ^{2} \right)$ 이므로 $\displaystyle \left( {\overline{\mathrm{AB}}} + {\overline{\mathrm{BF}}} \right) \times {\overline{\mathrm{SD}}} ^{2}$$= \left( a + b \right) \times 2 \left( a ^{2} - ab + b ^{2} \right)$ $\displaystyle = 2 \left( a ^{3} + b ^{3} \right) = \frac{35}{4}$ 에서 $\displaystyle a ^{3} + b ^{3} = \frac{35}{8}$ $\displaystyle V _{1} = {\overline{\mathrm{AB}}} \times {\overline{\mathrm{BC}}} \times {\overline{\mathrm{BF}}} = a ^{2} b$, $\displaystyle V _{2} = {\overline{\mathrm{AB}}} \times {\overline{\mathrm{BQ}}} \times {\overline{\mathrm{BF}}} = ab ^{2}$ 이므로 $\displaystyle V _{1} + V _{2} = a ^{2} b + ab ^{2} = \frac{15}{4}$ $\displaystyle \left( {\overline{\mathrm{AB}}} + {\overline{\mathrm{BF}}} \right) ^{3}$$= \left( a + b \right) ^{3}$ $= a ^{3} + 3 a ^{2} b + 3 ab ^{2} + b ^{3}$ $= a ^{3} + b ^{3} + 3 \left( a ^{2} b + ab ^{2} \right)$ $\displaystyle = \frac{35}{8} + \frac{45}{4} = \frac{125}{8}$ 이므로 $p = 8$, $q = 125$ 따라서 $p + q = 133$