It is required to construct a big rectangular hall to accommodate 500 persons, allowing 22.5 m³ space per person. The height of the hall is to be kept at 7.5 m, while the total inner surface area of the walls must be 1200 sq. m. Then the length and breadth of the hall respectively are

### Concept & System of Equations in Geometry This problem creates a system of two equations based on volume capacity and lateral surface area. $$ Total\_Volume = Number\_of\_Persons \times Volume\_per\_Person $$ $$ Area\_of\_Walls = 2H(L + B) $$ ### Step-by-Step Solution 1. **Find Total Required Volume**: - Volume = 500 persons × 22.5 m³/person = 11250 m³ 2. **Form Equation 1 (Volume)**: - Volume = L × B × H - 11250 = L × B × 7.5 - L × B = 11250 / 7.5 = 1500 3. **Form Equation 2 (Area of Walls)**: - Area = 2 × 7.5 × (L + B) = 15(L + B) - We are given Area = 1200 sq. m. - 15(L + B) = 1200 - L + B = 80 4. **Solve for L and B**: - We need two numbers that add up to 80 and multiply to 1500. - The numbers are 50 and 30. ### Exam Strategy & Shortcut Once you deduce that L + B = 80 and L × B = 1500, immediately scan the options. (a) 40+30=70 (Reject) (b) 45+35=80, but 45×35 = 1575 (Reject) (c) 50+30=80, and 50×30 = 1500 (Accept) (d) 60+20=80, but 60×20 = 1200 (Reject) ### Common Pitfall Using the total surface area formula $2(LB + BH + HL)$ instead of the area of the four walls $2H(L + B)$. "Total inner surface area of the walls" strictly excludes the floor and ceiling. ### Final Answer Therefore, the correct answer is **50 m and 30 m**.