Question

A model of a house is made on a scale of 1:100. If the height of the model is 12 cm, the area of the floor is 120 sq.cm. and the capacity of the house is 700 cu.cm. Then find the actual capacity of the house.
(a) 700 cu.m
(b) 300 cu.m
(c) 500 cu.m
(d) 900 cu.m

Answer 1

Hint: To solve the given question, we will first find out what a scale is and what the dimensions of the actual solid when the ratio is less than 1 will be. Then we will find the total volume of the house model considering the house to be cuboid. Then, we will find the volume of the house model which is not used. Let it be denoted by k. k will be obtained by subtracting the capacity of the house model from the volume of the cuboid. Then we will find the actual volume of the cuboid and the value of k by multiplying them with \[100\times 100\times 100.\] We will subtract the value of k from the actual volume of the cuboid to get the actual capacity.

Complete step by step solution:
Before we solve the given question, we must know what a scale is. A scale is a ratio of the dimension of the drawing/model to the dimension of the actual object. When scale ratio is less than one, the dimensions of the model are less than the dimensions of the actual object.
Now, let us consider that the total volume of the house model is x. Let the house be in the shape of a cuboid. We are given that the height is 12 cm and the area of the base is 120 sq.cm. The volume of the cuboid is the product of the height and base area. Thus,
\[x=12cm\times 120sq.cm\]
\[\Rightarrow x=1440cu.cm\]
Now, we will assume that the volume of the house model which is not used is k. Now, we know that the total volume of the house model will be the sum of the capacity of the house model and the volume of the house model which is not used. Thus,
\[x=k+\text{ capacity of model}\]
\[\Rightarrow 1440cu.cm=k+700cu.cm\]
\[\Rightarrow k=740cu.cm\]
Now, let the actual capacity of the house model be C. Now, we are given that the house is made on a scale of 1:100. Thus, the actual volume of the cuboid will be \[x\times 100\times 100\times 100cu.cm.\] Similarly, the actual volume of the house which is not used will be \[k\times 100\times 100\times 100cu.cm.\] Thus, we can say that,
The actual volume of the cuboid = Actual volume of the house not used + Actual capacity
\[\Rightarrow x\times 100\times 100\times 100=k\times 100\times 100\times 100+C\]
\[\Rightarrow C=\left( x-k \right)\times 100\times 100\times 100cu.cm\]
\[\Rightarrow C=\left( 1440-740 \right)\times 100\times 100\times 100cu.cm\]
\[\Rightarrow C=700\times 100\times 100\times 100cu.cm\]
Now, we know that,
1m = 100 cm
\[\Rightarrow 1m\times 1m\times 1m=100\times 100\times 100cu.cm\]
\[\Rightarrow 1cu.m=100\times 100\times 100cu.cm\]
Thus, we will get,
\[\Rightarrow C=700\times 1cu.m\]
\[\Rightarrow C=700cu.m\]
Hence, option (a) is the right answer.

Note: We can also calculate the actual capacity of the house without using the help of the base area and the height of the house model. Now, we will find the ratio of the capacity of the house model to the capacity of the actual house.
\[\dfrac{\text{Capacity of house model}}{\text{Capacity of actual house}}=\dfrac{1}{100}\times \dfrac{1}{100}\times \dfrac{1}{100}\left[ \text{because ratio of length is }\dfrac{1}{100} \right]\]
\[\Rightarrow \dfrac{700}{\text{Actual Capacity}}=\dfrac{1}{100\times 100\times 100}\]
\[\Rightarrow \text{Actual Capacity}=700\times 100\times 100\times 100cu.cm\]
\[\Rightarrow \text{Actual Capacity}=700cu.m\]