How to Solve Word Problems with Scale Drawings and Scale Factors
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Scale drawings are used in various fields such as architecture and engineering to represent larger objects on a manageable scale. A scale factor is used in scale drawings to determine the actual measurements of the object. Solving word problems involving scale drawings and scale factors involves several steps:
Step 1: Understand the Problem
The first step in solving any word problem is to understand what the problem is asking. Read the problem carefully, note down all given measurements and what you need to find.
Step 2: Identify the Scale Factor
Scale factor is the ratio of the length on the drawing to the actual length of the object. It is usually mentioned in the problem. If not, it can be calculated by dividing the length on the drawing by the actual length of the object.
Step 3: Apply the Scale Factor
If you are given the actual size and asked to find the size on the drawing, you multiply the actual size by the scale factor. If you are given the size on the drawing and asked to find the actual size, you divide the size on the drawing by the scale factor.
Step 4: Solve the Problem
Once you have the scale factor, and the measurements, you can perform the necessary calculations to find the missing measurements.
Example
John has a scale model of a building. The model has a height of 20 cm and the actual building has a height of 50 m. What is the scale factor of the model?
Solution:
First, we need to convert the actual height to the same units as the model. So, 50 m = 5000 cm.
Then, we find the scale factor using the formula:
\[ \text{Scale Factor} = \frac{\text{Size on Model}}{\text{Actual Size}} \]
Substituting the given values:
\[ \text{Scale Factor} = \frac{20}{5000} = 0.004 \]
So, the scale factor of the model is 0.004.
Exercises
1) A map has a scale of 1:50000. If 2 cm on the map represents an actual distance of 1 km, what is the actual distance of 5 cm on the map?
2) A model car is made to a scale of 1:18. If the actual car is 4.5 m long, how long is the model car?
3) The scale on a map is 1:250000. If a town is 5 cm apart from a city on the map, what is the actual distance in kilometers between the town and the city?
4) The blueprints for a house use a scale of 1:100. If a wall in the blueprint is 3 cm long, how long will the actual wall be?
5) A miniature model of a building is built to a scale of 1:500. If the model is 18 cm high, how tall is the actual building?
6) A scale model of a ship is built at a scale of 1:200. If the model ship is 1.5 m long, how long is the actual ship?
7) The scale on a map is 1:10000. If an actual area of land is 2 km², what area does it cover on the map?
8) A scale drawing of a room is drawn at a scale of 1:50. If a piece of furniture in the drawing measures 1.2 cm, how long is the actual piece of furniture?
9) A model of a skyscraper is built to a scale of 1:400. If the model is 20 cm tall, how tall is the actual skyscraper?
10) The scale on a map is 1:5000. If a lake is 3.5 cm wide on the map, what is the actual width of the lake?
1) The actual distance for 5 cm on the map is \(5 \times 50000 = 250000\) cm, or 2.5 km.
2) The model car is \(\frac{4.5}{18} = 0.25\) m long.
3) The actual distance between the town and the city is \(5 \times 250000 = 1250000\) cm, or 12.5 km.
4) The actual wall will be \(3 \times 100 = 300\) cm, or 3 m long.
5) The actual building is \(18 \times 500 = 9000\) cm, or 90 m tall.
6) The actual ship is \(1.5 \times 200 = 300\) m long.
7) The actual area of land covers \(2 \times 10000 = 20000\) cm² on the map.
8) The actual piece of furniture is \(1.2 \times 50 = 60\) cm long.
9) The actual skyscraper is \(20 \times 400 = 8000\) cm, or 80 m tall.
10) The actual width of the lake is \(3.5 \times 5000 = 17500\) cm, or 175 m.
