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The dimensions of a room are $\left( 9m\times 8m\times 6.5m \right)$ . It has one door of dimensions \[\left( 2m\times 1.5m \right)\] and two windows, each of dimensions \[\left( 1.5m\times 1m \right)\] . Find the cost of whitewashing the walls at $Rs.25$ per square meter.

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Last updated date: 16th Jul 2024
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Answer
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Hint: For solving this problem we first calculate the total area of four walls using the dimensions $\left( 9m\times 8m\times 6.5m \right)$ and surface area of the windows and door from \[\left( 1.5m\times 1m \right)\] and \[\left( 2m\times 1.5m \right)\] respectively. Upon calculating all the values of the area, we subtract the area of the door and the two windows from the total area of four walls to get the actual surface area to be whitewashed. Then, we multiply the cost of whitewashing per square meter with the final surface area to get the overall cost of whitewashing.

Complete step by step answer:
The dimensions of the room are $\left( 9m\times 8m\times 6.5m \right)$
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Hence, for the room \[Length=9m\] ; \[Breadth=8m\] and \[Height=6.5m\]
We know that the surface area of four walls of the room is \[2\left( length+breadth \right)\times height\]
We now substitute the values of height, breadth and length in the above expression and get
\[=2\left( 9+8 \right)\times 6.5\]
On further simplification we get the total surface area of the four walls of the room as
 \[=\text{ }34\text{ }\times \text{ }6.5\]
\[=221\text{ }{{\text{m}}^{\text{2}}}\]
 The dimensions of the door are \[\left( 2m\times 1.5m \right)\]
Hence, \[Length=2m\] and \[Breadth=1.5m\]
We know that area of one door is $length\times breadth$
We now substitute the values of breadth and length in the above expression and get
\[=~2\times 1.5\]
\[=~3\text{ }{{\text{m}}^{2}}\]
The dimensions of the windows are \[\left( 1.5m\times 1m \right)\]
Hence, for windows \[Length=1.5\text{m}\] and \[Breadth=1\text{m}\]
We know that area of two windows is \[2\left( length\times breadth \right)\]
We now substitute the values of breadth and length in the above expression and get
$=2\cdot (1.5\times 1)$
Further simplifying we get the area of two windows as $3\text{ }{{\text{m}}^{2}}$
The surface area to be whitewashed
\[=\text{Area of four walls of the room}\text{Area of one door}\text{Area of two windows}\]
By substituting the values in the above equation, we get
$=221-3-3$
$=215\text{ }{{\text{m}}^{2}}$
It is given that the cost of whitewashing is $Rs.25$ per metre square
Hence, total cost of whitewashing is $25\times 215=5375$ in Rupees.
Therefore, we conclude that the total cost of whitewashing is $Rs.5375$ .

Note: We must be careful that the surfaces to be whitewashed are only the walls. So, we must exclude the surfaces of the ceiling and floor of the room. Also, the room has two windows while calculating the area to be whitewashed. We should subtract the area of the window twice to get the correct area to be whitewashed.