A hall, whose length is $16\text{ m}$ and the breadth is twice its height, takes $168\text{ m}$ of paper with $2\text{ m}$ as its width to cover its four walls. The area of the floor is

### Concept & Surface Area Mapping The area of the paper required to cover the walls is exactly equal to the surface area of the four walls of the hall. $$ \text{Area of Paper} = \text{Length of Paper} \times \text{Width of Paper} = 2(l + b)h $$ ### Step-by-Step Solution * Calculate the total area of the wallpaper: $168\text{ m} \times 2\text{ m} = 336 \text{ m}^2$. * Let the height of the hall be $h$. The problem states breadth $b = 2h$. The length $l = 16\text{ m}$. * Set the area of the four walls equal to the paper area: $2(16 + 2h)h = 336$. * Divide the equation by 2: $(16 + 2h)h = 168$. * Expand the terms: $16h + 2h^2 = 168$. * Form a quadratic equation by dividing by 2 and moving all terms to one side: $h^2 + 8h - 84 = 0$. * Factor the quadratic equation: $(h + 14)(h - 6) = 0$. * Since height cannot be negative, $h = 6\text{ m}$. * Substitute $h$ back to find breadth: $b = 2(6) = 12\text{ m}$. * Calculate the floor area: $l \times b = 16 \times 12 = 192 \text{ m}^2$. ### Exam Strategy & Shortcut Instead of factoring the quadratic conventionally, you can test values directly from $(8 + h)h = 84$. Since $8 \times 10$ is $80$, $h$ must be slightly less than $10$. Testing $h=6$ immediately yields $(8 + 6) \times 6 = 14 \times 6 = 84$. ### Common Pitfall A standard trap is solving for $h$ ($6\text{ m}$) or $b$ ($12\text{ m}$) and mistakenly selecting it as the final answer instead of completing the last step to calculate the area of the floor. ### Final Answer Therefore, the correct answer is **$192 \text{ m}^2$**.