Question

A car manufacturer sells a similar, scale model of one of its real cars.
(a) The fuel tank of the real car has a volume of 64 litres and the fuel tank of the model has a volume of 0.125 litres.
Show that the length of the real car is 8 times the length of the model car.

Answer 1

Hint:This question is based on the constant ratio possible between the dimension and volume of the two geometrical shapes. We have to compute and then establish the relationship between the length of the real and model car with the volume of the real and model car. This is because the model car is the scale model of the real car.

Complete step-by-step answer:
Since the fuel tank is of cubic shape in general. Therefore we will compute the length of tanks of real and model cars by using the volume formula of a cube.
Since volume of cube = ${(side)^3}$
Thus,
Volume of real car,
\[
  {V_{real}} = {l_{real}}^3 \\
   \Rightarrow {l_{real}} = \sqrt[3]{{{V_{real}}}} \\
 \]
So, \[
  {l_{real}} = \sqrt[3]{{64}} \\
   \Rightarrow {l_{real}} = 4 \\
 \]
Now Volume of model car,
 \[
  {V_{model}} = {l_{model}}^3 \\
   \Rightarrow {l_{model}} = \sqrt[3]{{{V_{model}}}} \\
 \]
So, \[
  {l_{model}} = \sqrt[3]{{0.125}} \\
   \Rightarrow {l_{model}} = 0.5 \\
 \]
Now, ratio of length of real car and model car will be in same ratio as of length of fuel tank of real car and fuel tank of model car.
Now $
  \dfrac{{{l_{real}}}}{{{l_{model}}}} = \dfrac{4}{{0.5}} \\
   \Rightarrow {l_{real}} = 8 \times {l_{model}} \\
 $
Thus the length of the real car will be 8 times the length of the model car. Hence we have the proof.

Note:This problem is a good example of the simulation and modelling principles used in manufacturing science. With the help of a mathematical base, many problems for measurements can be solved easily.