Question

A model of a ship is made to a scale of 1:300. The length of the model of the ship is 2m. Calculate the length of the ship. The area of the deck of the ship is $ 180,000{{m}^{2}} $, Calculate the area of the deck of the model. The volume of the model is $ 6.5{{m}^{3}} $. Calculate the Volume of the ship.

Answer 1

Hint: Now the question can be solved with the knowledge of ratio and proportion. The ratio of the given quantities is the scale factor. Also if k is the scale factor length of model = k times length of the ship. Similarly Area of deck of model = $ {{k}^{2}} $ times area of deck of ship and Volume of model = $ {{k}^{3}} $ times volume of ship

Complete step-by-step answer:
Now we are given that the model of a ship is made to a scale 1:300 hence the scale factor is $\dfrac{1}{300}$.
Let the scale factor be k. hence, we have $k=\dfrac{1}{300}$ .
Now we know that length of the model = k times length of the ship.
The length of the model is given 2m.
$\begin{align}
  & \Rightarrow 2=\dfrac{1}{300}\times \text{ length of ship} \\
 & \Rightarrow 2\times 300\text{ }=\text{ length of ship} \\
 & \Rightarrow \text{ }600\text{ }=\text{ length of ship} \\
\end{align}$
Hence we get the length of the ship is 600m. ………………..(1)
Now let us proceed to find the Area of the model.
Since Area is a 2-dimensional quantity we will take the square of the scale factor.
Now Area of deck model = ${{k}^{2}}$ times area of deck ship.
$\begin{align}
  & \text{Area of deck of model = }{{\left( \dfrac{1}{300} \right)}^{2}}\times 180,000 \\
 & =\dfrac{1}{300}\times \dfrac{1}{300}\times 180,000 \\
 & =\dfrac{1}{300}\times 600 \\
 & =2 \\
\end{align}$
Hence we get Area of the deck of model is $2{{m}^{2}}$………(2)
Now in a similar way let us work for the volume of Ship.
Since Volume is a 3-dimensional quantity we will take a cube of the scale factor.
Now Volume of model = ${{k}^{2}}$ times Volume of ship.
$\begin{align}
  &\Rightarrow \text{6}\text{.5 = }{{\left( \dfrac{1}{300} \right)}^{3}}\times \text{ Volume of ship} \\
 &\Rightarrow 6.5=\dfrac{1}{300}\times \dfrac{1}{300}\times \dfrac{1}{300}\times \text{ Volume of Ship} \\
 &\Rightarrow 6.5\times 300\times 300\times 300=\text{ Volume of Ship}\text{.} \\
 &\Rightarrow \text{6}\text{.5}\times \text{27}\times \text{1}{{\text{0}}^{6}}=\text{ Volume of Ship} \\
 &\Rightarrow \text{65}\times \text{27}\times \text{1}{{\text{0}}^{5}}=\text{ Volume of Ship} \\
 &\Rightarrow \text{1755}\times \text{1}{{\text{0}}^{5}}=\text{ Volume of Ship}
\end{align}$
Hence we get Volume of Ship is $1755\times {{10}^{5}}{{m}^{3}}$ ………………….(3)
Hence from equation (1), equation (2), and equation (3) we have
Length of ship is 300m, Volume of deck of model is $2{{m}^{2}}$ and Volume of Ship is $1755\times {{10}^{5}}{{m}^{3}}$

Note: Now we multiply the scale factor according to the dimension of the quantity taken hence while taking Area and Volume take the square and cube of the scale factor respectively. Also while writing the scale factor make sure it is written properly For example if the model of a ship is made to a scale of 1:300 this means that scale factor is $ \dfrac{1}{300} $ and the model is $ \dfrac{1}{300} $ times the Ship. Do not be mistaken by writing the Ship is $ \dfrac{1}{300} $ times the model.