Scale factor length, surface area and volume (4 worded problems)

Scale Factor Length, Surface Area and Volume Problems

Problem One: Garden Designing

John, a garden designer, is working on a 10m by 20m garden, but his planning is based on a 2m by 4m model. Here, the scale factor of length is 5 (10m/2m = 5).
Therefore, all the dimensions in the actual garden are 5 times larger than the model.
For the surface area, the scale factor becomes 5² = 25 times the model, and for the volume, the scale factor is 5³ = 125 times the model.

Problem Two: Baking

Emma wants to make a larger version of her favourite 5cm by 5cm by 5cm cake. The new cake needs to be 15cm by 15cm by 15cm. The scale factor of length is 3 (15cm/5cm = 3).
So, the dimensions in the new cake are 3 times larger than the old cake.
For surface area, the scale factor is 3² = 9 times the old cake, and for volume, the scale factor is 3³ = 27 times the old cake.

Problem Three: Map Reading

A 5cm square on a map represents a real 500m square area. So, the scale factor of length is 10,000 (500m/0.05m = 10,000).
Therefore, each meter in the real area is represented by 0.0001m (or 0.1mm) on the map.
For surface area, the scale factor is (10,000)² = 100,000,000, meaning each square meter in the real area is represented by 0.00000001 square meters on the map.

Problem Four: Architectural Designing

An architect has a blueprint of a 1m by 1m room but wants to build a 4m by 4m version. The scale factor of length will be 4 (4m/1m = 4).
Thus, each dimension in the actual room is 4 times larger than the blueprint.
For surface area, the scale factor is 4² = 16 times the blueprint, and for volume, the scale factor is 4³ = 64 times the blueprint.

Remember to revise these problems and apply the principles of scale factors to similar real-world problems.