Question

A room is \[9\] m by \[8\] m by \[6.5\] m. It has one door or dimension \[\left( {2m \times 1.5m} \right)\] and four windows each of dimension \[\left( {1.5m \times 1m} \right)\]. Find the cost of painting the walls at Rs. 50 per m2.

Answer 1

Hint:
Here we will find the total surface area of the room by substituting the value of the dimension of the room in the total surface area formula. Then will find the area of the door and the two windows by using their dimension and then subtract the area of door and window from the total surface area. Finally, we will multiply the new surface area with the rate of whitewashing the walls to find the cost.

Formula used:
We will use the following formulas:
1) Total surface area of room \[ = 2\left( {lb + lh + bh} \right){m^2}\], where, \[l = \] Length of the room, \[b = \] Breadth of the room, \[h = \] Height of the room
2) Area of door \[ = \left( {l \times b} \right){m^2}\], where, \[l = \] Length of the door, \[b = \] Breadth of the door
3) Area of window \[ = \left( {l \times b} \right){m^2}\], where, \[l = \] Length of the window, \[b = \] Breadth of the window

Complete step by step solution:
The dimension of the room is given as:
\[l = 9m\]
\[b = 8m\]
\[h = 6.5m\]
We will now find the total surface area of the room.
Substituting all the value in the formula Total surface area of room \[ = 2\left( {lb + lh + bh} \right){m^2}\], we get
Total surface area of room \[ = 2\left( {9 \times 8 + 9 \times 6.5 + 8 \times 6.5} \right){m^2}\]
Multiplying the terms, we get
\[ \Rightarrow \] Total surface area of room \[ = 2\left( {72 + 58.5 + 52} \right){m^2}\]
Adding the terms, we get
\[ \Rightarrow \] Total surface area of room \[ = 2 \times 182.5\]
Multiplying the terms, we get
\[ \Rightarrow \] Total surface area of room \[ = 365{m^2}\]
So, our Total surface area is \[365{m^2}\].
Next, we will find the area of doors and windows.
Dimension of the door is given as \[l = 2m\] and \[b = 1.5m\].
Substituting above values in the equation Area of door \[ = \left( {l \times b} \right){m^2}\], we get
Area of door \[ = \left( {2 \times 1.5} \right){m^2}\]
Multiplying the terms, we get
\[ \Rightarrow \] Area of door\[ = 3{m^2}\]
Area of window \[ = \left( {l \times b} \right){m^2}\]…..\[\left( 3 \right)\]
Dimension of the window is given as \[l = 1.5m\] and \[b = 1m\].
Substituting above values in the equation Area of door \[ = \left( {l \times b} \right){m^2}\], we get
Area of one window \[ = \left( {1.5 \times 1} \right){m^2}\]
Multiplying the terms, we get
\[ \Rightarrow \] Area of window \[ = 1.5{m^2}\]
As the room has two windows therefore area of window will be
Area of two windows \[ = 2 \times 1.5{m^2}\]
Multiplying the terms, we get
\[ \Rightarrow \] Area of two windows \[ = 3{m^2}\]
Now we will calculate the surface area of the wall.
Putting all value in above formula Surface area of wall \[ = \] Total surface area \[ - \] Area of door \[ - \] Area of windows, we get,
Surface area of wall \[ = 365 - 3 - 3\]
Adding and subtracting the terms, we get
\[ \Rightarrow \] Surface area of wall \[ = 359{m^2}\]
So, our Surface area of the wall is \[359{m^2}\].
Now, it’s given that cost of whitewashing the wall is \[{\rm{Rs}}.50{m^2}\]
Cost of painting the wall \[ = 50 \times 359\]
Multiplying the terms, we get
\[ \Rightarrow \] Cost of painting the wall \[ = {\rm{Rs}}.17950\]

So, the cost of painting the walls is Rs. 17950.

Note:
Here, in order to find the total surface area of the whole room which is to be painted, we will subtract the area of the things that are not to be painted. It is important for us to check whether the units of all the dimensions are the same or not. If they are not the same then we have to make them the same otherwise we will get wrong answers.