Question

A model of a ship is made to a scale of $1:250$. Calculate
(i)The length of the ship, if the length of the model is 1.6 m
(ii)The area of the deck of the ship, if the area of the model is 2.4 ${{\text{m}}^2}$.
(iii)The volume of the model, if the volume of the ship is 1 ${\text{k}}{{\text{m}}^3}$.

Answer 1

Hint: We will express the ratio of length of the model and the ship as a fraction and equate it to the given ratio. For the area, we will equate the ratio of areas to the square of the given ratio. Finally, for the volume we will express the ratio of volumes to the cube of the given ratio.

Complete step-by-step answer:
Let us assume Dimension of model : Dimension of ship = $1:250$
(i)Length of model : Length of ship = $1:250$
$\dfrac{{{\text{length of model}}}}{{{\text{length of ship}}}} = \dfrac{1}{{250}}$ ……………………………………(1)
It is given that the length of the model is \[1.6\] m.
Substituting this in equation (1), we get
$\dfrac{{1.6}}{{{\text{length of ship}}}} = \dfrac{1}{{250}}$. Therefore, length of ship $ = 1.6 \times 250 = 400$ m

(ii)Area of model : Area of ship $ = {1^2}:{250^2}$
i.e., $\dfrac{{{\text{area of model}}}}{{{\text{area of ship}}}} = \dfrac{{{1^2}}}{{{{250}^2}}}$ ………………………(2)
It is given that the area of the model is 2.4 ${{\text{m}}^2}$.
Substituting this in equation (2), we get
$\dfrac{{2.4}}{{{\text{area of ship}}}} = \dfrac{{{1^2}}}{{{{250}^2}}}$.
Therefore, the area of the deck of the ship $ = 2.4 \times {250^2} = 2.4 \times 62500 = 150000$${{\text{m}}^2}$

(iii)Volume of model : Volume of ship $ = {1^3}:{250^3}$
i.e., $\dfrac{{{\text{volume of model}}}}{{{\text{volume of ship}}}} = \dfrac{{{1^3}}}{{{{250}^3}}}$ ………………………..(3)
It is given that the volume of the ship ${\text{ = 1 k}}{{\text{m}}^3}$.
We know that 1 km \[=\] 1000 m.
Thus, volume of ship $ = 1{\text{ k}}{{\text{m}}^3} = {(1000{\text{ m}})^3} = {\left( {1000} \right)^3}{\text{ }}{{\text{m}}^3}$.
Substituting this in equation (3), we get
$\dfrac{{{\text{volume of model}}}}{{{{1000}^3}}} = \dfrac{{{1^3}}}{{{{250}^3}}}$.
Therefore, the volume of the model $ = \dfrac{{{{1000}^3}}}{{{{250}^3}}} = {\left( {\dfrac{{1000}}{{250}}} \right)^3} = {4^3} = 64{\text{ }}{{\text{m}}^3}$


Note: Since length is a one-dimensional quantity, we have retained the ratio as such. On the other hand, area and volume are two-dimensional and three-dimensional quantities respectively. This is the reason we have squared and cubed the ratios accordingly for area and volume. Area is defined as the space occupied by a two dimensional figure whereas volume is defined as the capacity of a three dimensional object.