Enlargements/Similar shapes - Intermediate & Higher tier - WJECFurther examples for Higher tier

A company manufactures boxes and two of them are shown below. The boxes are similar.

1. Calculate the dimensions of box 2
2. By considering the area of each face, calculate the surface area of box 1
3. Without calculating areas of each face, ie using scale factors, calculate the surface area of box 2
4. Calculate the volume of box 1
5. Without calculating the volume by multiplying lengths, ie using scale factors, calculate the volume of box 2
6. A carrier transports the boxes at a charge of £1.84 for box 1 and charges are dependent upon volume. How much will the carrier charge for transporting box 2?

Solution

1. Scale factor = 3.5 ÷ 1.4 = 2.5. So dimensions of box 2 are 3.5 cm by 12.5 cm by 20 cm
2. Surface area of box 1 = (2 × 1.4 × 5) + (2 × 1.4 × 8) + (2 × 5 × 8) = 116.4 cm²
3. Area scale factor = 2.52 = 6.25. So surface area of box 2 = 6.25 × 116.4 = 727.5 cm²
4. Volume of box 1 = 1.4 × 5 × 8 = 56 cm³
5. Volume scale factor = 2.5³ = 15.625. So volume of box 2 = 15.625 × 56 = 875 cm³
6. Volume scale factor = 15.625. So transport costs for box 2 = 15.625 × £1.84 = £28.75

Example 2

The Big Stopper Company manufactures gobstoppers. Three of its range are:

• the midi with a diameter of 2 cm
• the large with a diameter of 3.2 cm and
• the giant which has a diameter of 5 cm

The company claims that the large is over four times the size of the midi and that the giant is four times the size of the large. Are these claims true?

The midi costs 8p to manufacture. Assuming that costs per cm³ are the same for all gobstoppers, how much does it cost to manufacture the other two? Work to the nearest penny where appropriate.

Solution

Large gobstopper compared to midi:

Scale factor for length = 3.2 ÷ 2 = 1.6

Scale factor for volume = 1.6³ = 4.096

The company is correct in saying that the large is over four times the size (or volume) of the midi but it is only slightly over four times. It would be better to say that it is four times the size.

Giant gobstopper compared to large:

Scale factor for length = 5 ÷ 3.2 = 1.5625

Scale factor for volume = 1.5625³ = 3.814697266 = 3.81 (to two decimal places).

The company is not correct in saying that the giant is four times the size (or volume) of the large as it is only 3.81 times the volume. They would be better to say that it is nearly four times the size, or well over three times the size.

Manufacturing cost of large gobstopper = 4.096 × 8p = 32.768 p = 33p (nearest penny).

Manufacturing cost of giant gobstopper = 3.814697266 × 32.768 p = 125p = £1.25