Question

The dimension of the model of a multi storey building is 1 m by 60 cm by 1.20 m, if the scale factor is 1:50, find the actual dimension of the building. Also find the floor area of a room of the building, if the floor area of the corresponding room in the model is 50 sq.cm.

Answer 1

Hint: It is given that the scale factor is 1:50. It means that a unit distance in the model represents 50 times of unit distance in the actual building. Now, multiply by the scale factor 50 in the given dimensions of the building to get the dimensions of the actual building. The floor area of the corresponding room in the model is 50 sq.cm. To find the actual floor area, multiply by the scale factor twice in the given floor area of the corresponding room of the model. We know that \[100cm=1m\] and \[10000c{{m}^{2}}=1{{m}^{2}}\] . Now, solve it further and get the dimensions of the building and the area of the corresponding room in the model.

Complete step-by-step answer:
According to the question, it is given that
The length of the multi storey building = 1 m …………………………………….(1)
The breadth of the multi storey building = 60 cm …………………………………….(2)
The height of the multi storey building = 1.20 m …………………………………….(3)
The floor area of the corresponding room in the model = 50 sq.cm = \[50\,cm\times cm\] …………………………………..(4)
It is also given that the scale factor is 1:50. It means that a unit distance in the model represents 50 times of unit distance in the actual building ………………………………. (5)
Now, from equation (1) and equation (5), and multiplying by 50 in equation (1), we get
The length of the multi storey building = \[1\times 50\] m = 50 m …………………………………(6)
Now, from equation (2) and equation (5), and multiplying by 50 in equation (2), we get
The breadth of the multi storey building = \[60\times 50\] cm = \[30\times 100\] cm …………………………………(7)
We know that \[100cm=1m\] ………………………….(8)
Now, from equation (7) and equation (8), we get
The breadth of the multi storey building = 30 m …………………………………..(9)
Now, from equation (3) and equation (5), and multiplying by 50 in equation (3), we get
The height of the multi storey building = \[1.20\times 50\] m = 60 m …………………..………………(10)
From equation (5), equation (9), and equation (10), we have the dimensions of the actual building.
The length, breadth, and height of the actual building are 50 meters, 30 meters, and 60 meters respectively
Similarly, from equation (4), we have the floor area of the corresponding room in the model = 50 sq.cm = \[50\,cm\times cm\]
 Now, from equation (4) and equation (5), we get
The floor area of the corresponding room in the model = \[50\times {{\left( 50cm \right)}^{2}}=125000c{{m}^{2}}\] ……………………………………(11)
We also know that \[10000c{{m}^{2}}=1{{m}^{2}}\] ……………………………….(12)
Now, from equation (11) and equation (12), we get
The floor area of the corresponding room in the model = \[12.5\times 10000c{{m}^{2}}=12.5{{m}^{2}}\] …………………………………..(13)
Therefore, the dimensions of the actual building are 50 meters by 30 meters by 60 meters, and the floor area of the corresponding room in the model is \[12.5{{m}^{2}}\] .

Note: Since the given scale factor is 1:50 here, one might make a mistake in multiplying the ratio factor. One might multiply by 50 in the given area of the floor and conclude it as the area of the actual floor. This is wrong because a unit length in the model represents 50 times of unit length in the actual building and in the unit of the area we have the square of the length. So, we have to multiply the scale factor twice. Hence, the area of the floor is \[50\times {{\left( 50cm \right)}^{2}}=125000c{{m}^{2}}\] .