Question

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 `2000cm^3` .
Is that right?

Answer 1

`V_("air in the actual car") = 5,000,000 cm^3`

Explanation:

Given: `V_("air in the model") = 40 cm^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_("air in the actual car") = 40 cm^3 xx 50^3 = 5,000,000 cm^3`

Answer 2

`"5,000,000"cm^3=5m^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 `50xx50=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 `50xx50xx50="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? `"125,000"`. So we end up with the original car having:

`40xx"125,000"="5,000,000"cm^3=5m^3` of air.

Answer 3

See below

Explanation:

If `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:

`(40 cm^3)/x= (1)^3/(50)^3`

`x= 40cm^3*50^3=5000000" cm"^3"`

To convert to meters cubed if you wanted:

`5000000 "cm"^3xx(1 m)^3/(100 cm)^3= 5 m^3`

Answer 4

`5m^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 xx 40 cm^3`

`x = 5,000,000 cm^3`

This is not a practical unit, convert to `m^3`

`(5,000,000)/(1000xx1000)`

`= 5m^3`