An open box is made of wood 3 cm thick. Its external dimensions are 1.46 m, 1.16 m and 8.3 dm. The cost of painting the inner surface of the box at 50 paise per 100 sq. cm is

### Concept & Inner Surface Area of an Open Box To find the inner surface area of an open box, you must calculate the area of the four inner walls plus the inner base. Formula: $$ \text{Inner Surface Area} = (l \times b) + 2(bh + hl) $$ ### Step-by-Step Solution 1. **Given:** External dimensions: $L = 1.46$ m $= 146$ cm, $B = 1.16$ m $= 116$ cm, $H = 8.3$ dm $= 83$ cm. Thickness $t = 3$ cm. 2. **Calculate Internal Dimensions:** Since it is an open box: $l = 146 - 2(3) = 140$ cm. $b = 116 - 2(3) = 110$ cm. $h = 83 - 3 = 80$ cm (no top thickness). 3. **Calculate Inner Surface Area:** Area of base = $140 \times 110 = 15400$ cm$^2$. Area of walls = $2(110 \times 80 + 80 \times 140) = 2(8800 + 11200) = 2(20000) = 40000$ cm$^2$. Total Inner Surface Area = $15400 + 40000 = 55400$ cm$^2$. 4. **Calculate Cost:** Rate is 50 paise (0.5 Rupees) per 100 cm$^2$. Total Cost = $\frac{55400}{100} \times 0.5 = 554 \times 0.5 = 277$ Rupees. ### Exam Strategy & Shortcut Ensure all units are converted to centimeters first to match the given painting rate of 100 sq. cm. Be careful to convert dm (decimeters) to cm correctly ($1 \text{ dm} = 10 \text{ cm}$). ### Common Pitfall Failing to unify the units before calculating the dimensions, particularly misinterpreting 8.3 dm. ### Final Answer Therefore, the correct answer is **₹ 277**.