The ratio of the height of a room to its semi-perimeter is $2 : 5$. It costs ₹ $260$ to paper the walls of the room with paper $50\text{ cm}$ wide at ₹ $2$ per metre allowing an area of $15\text{ sq. m}$ for doors and windows. The height of the room is

### Concept & Abstract Dimensions The area covered by the wallpaper can be reverse-calculated from its total cost and cost per unit length. The semi-perimeter is $(l + b)$. $$ \text{Total Wall Area} = \text{Area of Paper} + \text{Area of Doors/Windows} $$ ### Step-by-Step Solution * Given the ratio of height to semi-perimeter ($l+b$) is $2:5$. Let height $h = 2x$ and semi-perimeter $(l+b) = 5x$. * The width of the paper is $50\text{ cm} = 0.5\text{ m}$. The cost is ₹ $2$ per running metre. * The area of a $1\text{-metre}$ length of paper is $1\text{ m} \times 0.5\text{ m} = 0.5 \text{ m}^2$. Thus, the rate is ₹ $2$ for every $0.5 \text{ m}^2$, which translates to a rate of ₹ $4$ per $1 \text{ m}^2$. * Total cost for the paper is ₹ $260$. The total area of paper used = $260 / 4 = 65 \text{ m}^2$. * The total area of the four walls includes the papered area plus the doors and windows. Total Wall Area = $65 + 15 = 80 \text{ m}^2$. * We know the formula for wall area is $2(l+b)h$. Substitute our $x$ variables: $2(5x)(2x) = 80$. * Simplify the equation: $20x^2 = 80 \Rightarrow x^2 = 4 \Rightarrow x = 2$. * Calculate the height: $h = 2x = 2(2) = 4\text{ m}$. ### Exam Strategy & Shortcut Calculate the effective rate per square meter immediately. ₹$2$ for $0.5\text{ m}$ width means ₹$4$ per sq.m. The $260$ cost gives $65\text{ sq.m}$, plus $15$ equals $80\text{ sq.m}$. Knowing area is $2(5x)(2x) = 20x^2$, $20x^2=80$ provides $x=2$ in seconds. ### Common Pitfall A tricky element is confusing the "₹$2$ per metre" rate as a rate per square meter. Because the paper is only $0.5\text{ m}$ wide, $1$ running meter represents only $0.5$ square meters of material. ### Final Answer Therefore, the correct answer is **$4\text{ m}$**.