How big is a toy car, really?
Tonight's idea
A 1:64 model is a copy in which every length is 64 times shorter than the real thing. A real car 4.48 m long becomes \(448 \div 64 = 7\) cm. Height, width, wheel size: all divided by 64.
Flat things shrink more. A window has a width and a height. Both are divided by 64, so its area is divided by \(64 \times 64 = 4{,}096\).
Solid things shrink more again. A block has length, width and height. All three are divided by 64, so its volume is divided by \(64 \times 64 \times 64 = 262{,}144\).
Scale 1:64. Lengths ÷ 64. Areas ÷ 4,096. Volumes ÷ 262,144. See blueprints.
For mass, one more condition is needed: the model must be an exact scaled copy all the way through. Every solid part and every empty space must shrink in the same way, with matching parts made of the same material. Then mass follows volume, so the model is 262,144 times lighter. That is the surprise: only 64 times shorter, but more than a quarter of a million times lighter.
Watch one: shrink a 1,310 kg car
Length: \(448 \div 64 = 7\) cm.
Windscreen: the real one is 12,000 cm². \(12{,}000 \div 4{,}096 \approx 2.9\) cm², smaller than a postage stamp.
Mass: 1,310 kg is 1,310,000 g, and \(1{,}310{,}000 \div 262{,}144 \approx 5\) g. That is about one sheet of printer paper.
A real toy car is heavier than 5 g. It is not built like a real car: it is mostly solid metal, and a real car is mostly empty space.
Your turn
Every model here is 1:64. For Question 3 and the stretch, assume an exact scaled copy: every solid part and empty space is scaled in the same way, with matching parts made of the same material.
- (a) A real truck is 9.6 m long. How long is the model, in cm? (b) A model wheel is 1 cm across. How wide is the real wheel? (c) A model car is 7.5 cm long. How long is the real car, in metres?
- The bonnet (hood) of a model is a rectangle 2 cm by 1.5 cm. (a) Find its area. (b) Find the length and width of the real bonnet, then its area in cm². (c) Divide the real area by the model's area. What do you notice?
- (a) A real car has a mass of 1,500 kg. Find the model's mass in grams, to one decimal place. (b) A model of a bronze statue has a mass of 40 g. Find the mass of the real statue in tonnes. (1 tonne = 1,000,000 g.)
- For each, decide whether to divide by 64, by 4,096 or by 262,144. Then find the model's amount. (a) The real aerial is 80 cm long. (b) The real fuel tank holds 50,000 cm³. (c) The real car needs 4,000 cm³ of paint, and the model's coat of paint is just as thick.
Stretch: work back from the mass
Under the same exact-copy assumption, a model is 1,000 times lighter than the real thing. What is its scale? What scale makes a model a million times lighter?
The square-cube law. Double every length and cross-sectional areas grow \(2^2 = 4\) times, but volume and mass grow \(2^3 = 8\) times. That is one reason an elephant needs proportionally thicker legs than a mouse: weight grows faster than the area of bone supporting it.
Answers
- (a) \(960 \div 64 = 15\) cm. (b) 64 cm. (c) \(7.5 \times 64 = 480\) cm, which is 4.8 m.
- (a) \(2 \times 1.5 = 3\) cm². (b) \(2 \times 64 = 128\) cm by \(1.5 \times 64 = 96\) cm, area \(128 \times 96 = 12{,}288\) cm². (c) \(12{,}288 \div 3 = 4{,}096\), which is \(64 \times 64\).
- (a) \(1{,}500{,}000 \div 262{,}144 \approx 5.7\) g. (b) \(40 \times 262{,}144 = 10{,}485{,}760\) g, about 10.5 tonnes.
- (a) A length: \(80 \div 64 = 1.25\) cm. (b) A volume: \(50{,}000 \div 262{,}144 \approx 0.19\) cm³, a few drops. (c) Paint of the same thickness covers an area, so divide by 4,096: \(4{,}000 \div 4{,}096 \approx 0.98\) cm³, about 1 cm³.
Stretch. Mass is divided by the scale number three times. \(10 \times 10 \times 10 = 1{,}000\), so the scale is 1:10. \(100 \times 100 \times 100 = 1{,}000{,}000\), so a million times lighter means 1:100.