Question

A model of a ship is made to a scale of \[1:250\]. Calculate:
A) The length of the ship, if the length of model is \[1.6\;m\].
B) The area of the deck of the ship, if the area of the deck of model is \[2.4\;{m^2}\].
C) The volume of the model, if the volume of the ship is \[1\;k{m^3}\].

Answer 1

Hint: First we have to know the ratio of the given quantities is the scale factor. Lat \[x\] is the scale factor, then the length of the model of the ship is equal to \[x\] times the length of the ship. Similarly, Area of deck of model of a ship is equal to \[{x^2}\] times area of deck of ship and Volume of model of a ship is equal to \[{x^3}\] times volume of ship.

Complete step by step solution:
Since the given model of a ship is made to a scale of \[1:250\]. Then by definition of ratio we get the scale factor is
\[x = \dfrac{1}{{250}}\]-------(1)

(i) Given the length of model ship is \[1.6\;m\] ----(2)
Since length of the model of a ship \[ = \dfrac{1}{{250}}\] times the length of ship
Then length of the ship\[ = 250 \times 1.6\]
Hence, the length of the ship is \[400\;m\].

(ii) Given the area of the deck of the model is \[2.4\;{m^2}\].
Now, Area of deck of model \[ = {x^2}\] times area of deck of a ship
\[ \Rightarrow \] Area of deck of model \[ = {\left( {\dfrac{1}{{250}}} \right)^2}\] times area of deck of a ship
\[ \Rightarrow \] Area of the deck of a ship\[ = {\left( {250} \right)^2} \times 2.4\; = 150000\]
Hence the area of the deck of a ship is \[1,50,000\;{m^2}\].

(iii) Given the volume of the ship is \[1\;k{m^3} = {10^9}\;{m^3}\].
Now, Volume of model of a ship \[ = {x^3}\] times volume of a ship
\[ \Rightarrow \] Volume of model \[ = {\left( {\dfrac{1}{{250}}} \right)^3}\] times volume of a ship
\[ \Rightarrow \] Volume of model of a ship\[ = {\left( {\dfrac{1}{{250}}} \right)^3} \times {10^9}\;{m^3} = 64\;{m^3}\]
Hence volume of the model of a ship is \[64\;{m^3}\].

Note:
Note that area is a two-dimensional quantity so to find the area of the model of a ship, we will take the square of the scale factor. Similarly, volume is a three-dimensional quantity so to find the volume of the model of a ship, we will take the cube of the scale factor. The ratio of two quantities is the fraction of one quantity to another quantity.