Final answer:
A scale factor is a ratio that describes the proportional relationship between a model's dimensions and the actual object's dimensions. It is used to calculate actual sizes or areas from scale models or drawings by multiplying the scaled dimensions by the scale factor.
Step-by-step explanation:
Understanding Scale Factors
A scale factor is used in mathematics to describe the ratio of the dimensions of a drawing, map, or model to the actual dimensions of the object it represents. It is essential in problems involving similarity and geometric figures.
Let's address some exercises relating to scale factor:
- In the case of the donut problem, the scale factor from one donut to another could be expressed as 1:4 or as a fraction, which is 1/4. This means the one donut is four times smaller than the other donut.
- For a drawing with a scale factor of 1/200, to find the actual dimensions of an object, you would multiply the drawing's dimensions by 200. If you need the area, you would then calculate it normally using the actual dimensions.
- When Calvin drew his map with a scale factor of 1/800, and knowing the actual distance is 80 meters, you convert meters to centimeters (since 1 meter = 100 centimeters) and then divide by 800 to find the distance on the map. In this case, the distance on the map would be 10 centimeters.
- To determine the dimensions of a larger square that is twice the size of a smaller one, multiply the side length of the smaller square by 2. If the side length is 4 inches, the larger square's side length would be 8 inches.
- The scale factors for different ratios are calculated by dividing the model's dimension by the actual dimension. If the width of a small rectangle is 2 inches and the similar larger rectangle is 9 inches, the scale factor would be 2/9.
- For the drawing of a man 4 inches high, compared to the actual height of 64 inches, the scale factor for the drawing is 4/64 or 1/16.
- A map scale of 1 inch to 4 feet means that each inch on the map represents 4 feet in reality. Therefore, the scale factor is 1/48 because there are 12 inches in a foot (1 inch/4 feet = 1/4*12).
- If the drawing of a box has dimensions in inches, and the actual box is 3(1/4) times larger, you simply multiply each dimension by 3.25 to find the actual dimensions.
These principles allow you to comfortably move between scale models and actual sizes, understanding the proportional relationships thanks to the scale factor.