# The volume of air in a model car is 40cm^3 . A scale of 1:50 was used to create the model. What is the volume of air in the actual car?

## I got the answer $2000 c {m}^{3}$ . Is that right?

Jul 23, 2018

${V}_{\text{air in the actual car}} = 5 , 000 , 000 c {m}^{3}$

#### Explanation:

Given: ${V}_{\text{air in the model}} = 40 c {m}^{3}$. Scale of $1 : 50$ model to actual.

The scale $1 : 50$ is a linear scale. This means for each centimeter in length, width and height of the model, there is $50$ centimeters in length, width and height in the actual model.

The volume scale is then ${50}^{3}$

${V}_{\text{air in the actual car}} = 40 c {m}^{3} \times {50}^{3} = 5 , 000 , 000 c {m}^{3}$

$\text{5,000,000} c {m}^{3} = 5 {m}^{3}$

#### Explanation:

Let's think about the calculation this way:

Let's say we have a line segment and it's length 1. We then want to scale that line using the 1:50 ratio. Our scaled line segment will be length 50. Very straightforward calculation.

Let's now look at a square. It's got length of 1 and width of 1 and an area of 1 unit squared. Let's now apply the scale. What happens? The length is now 50 and the width is also 50. The area of the square is $50 \times 50 = 2500$ units squared.

The same thing happens when we move to three dimensions: our cube with length 1, width 1, height 1, and volume 1 unit cubed, when scaled, now has width 50, length 50, height 50 and volume $50 \times 50 \times 50 = \text{125,000}$ units cubed.

Let's now move to our question. We have a car that has a volume of 40 cm cubed. What factor do we use for scaling to the actual size of the car? $\text{125,000}$. So we end up with the original car having:

$40 \times \text{125,000"="5,000,000} c {m}^{3} = 5 {m}^{3}$ of air.

Jul 25, 2018

See below

#### Explanation:

If $\frac{1}{50}$ was a linear ratio, meaning that it was applied to each of the three dimensions of the car and not just the volume then:

$\frac{40 c {m}^{3}}{x} = {\left(1\right)}^{3} / {\left(50\right)}^{3}$

$x = 40 c {m}^{3} \cdot {50}^{3} = 5000000 \text{ cm"^3}$

To convert to meters cubed if you wanted:

$5000000 {\text{cm}}^{3} \times {\left(1 m\right)}^{3} / {\left(100 c m\right)}^{3} = 5 {m}^{3}$

Jul 26, 2018

$5 {m}^{3}$ is the volume of the real car.

#### Explanation:

It is important to realise that the model and the real car are similar figures.

They have exactly the same shape, but different sizes.
The scale is $1 : 50$ which means that all lengths on the real car are $50$ times bigger than on the model.

In similar figures, their volumes are in the same ratio as the cube of the lengths.

1^3/50^3 = (40cm^3)/x" "(larr "model")/(larr "real car")

$x = {50}^{3} \times 40 c {m}^{3}$

$x = 5 , 000 , 000 c {m}^{3}$

This is not a practical unit, convert to ${m}^{3}$

$\frac{5 , 000 , 000}{1000 \times 1000}$

$= 5 {m}^{3}$