Subjects geometry

Room Surface Area

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1. **State the problem:** We have a room with dimensions 5.8 m (length), 3.9 m (width), and 2.43 m (height). There are windows and doors with given dimensions. We need to find the total surface area inside the room, the total area of the floor, windows, and doors, and finally the wallpapered surface area. 2. **Calculate total surface area of the inside of the room:** The room is a rectangular prism, so the total surface area inside is the sum of the areas of all walls, the floor, and the ceiling. Formula for surface area of a rectangular prism: $$\text{Surface Area} = 2(lw + lh + wh)$$ where $l=5.8$, $w=3.9$, $h=2.43$. Calculate each term: $$lw = 5.8 \times 3.9 = 22.62$$ $$lh = 5.8 \times 2.43 = 14.094$$ $$wh = 3.9 \times 2.43 = 9.477$$ Sum: $$22.62 + 14.094 + 9.477 = 46.191$$ Total surface area: $$2 \times 46.191 = 92.382\, m^2$$ 3. **Calculate the area of the floor, windows, and doors:** - Floor area: $$5.8 \times 3.9 = 22.62\, m^2$$ - One window area: $$2.9 \times 1.1 = 3.19\, m^2$$ - Two windows area: $$2 \times (1.21 \times 1.1) = 2 \times 1.331 = 2.662\, m^2$$ - Two doors area: $$2 \times (1.3 \times 2.1) = 2 \times 2.73 = 5.46\, m^2$$ Sum of floor, windows, and doors: $$22.62 + 3.19 + 2.662 + 5.46 = 33.932\, m^2$$ 4. **Calculate the wallpapered surface area:** Wallpaper covers the total surface area minus the area of the floor, windows, and doors: $$\text{Wallpaper area} = 92.382 - 33.932 = 58.45\, m^2$$ Rounded to two decimal places: $$\boxed{58.45\, m^2}$$