Josh is so excited about Mount Everest that he has decided to create a model of the mountain for his geography class. Mr. Watkins approved the project and so Josh is putting together all of the information. During lunch, instead of talking with his friends, he took out his notebook and began figuring out how to measure the mountain. His friend Sasha spotted him and came over.
“Hi Josh, what are you working on?” she asked.
“Well, I am going to build a model of Mount Everest for my geography project. I am reading this really cool book about it. So I’m trying to figure out what scale to use for the model,” he explained grinning from ear to ear.
“What do you mean “scale”?”
“If you are going to build a model, you have to use a scale. You shrink down the real dimensions so that you can actually build a model that everyone will be able to see, think about it, Everest is 29,035 feet high, you can’t build that using the real dimensions,” Josh said.
“Oh I get it, so you use \begin{align*}1^{\prime\prime}\end{align*} for 1 foot.”
“That’s the right idea, but @$\begin{align*}1^{\prime\prime}\end{align*}@$ is definitely too big. If I used @$\begin{align*}1^{\prime\prime}\end{align*}@$ equals 1 foot for the scale, I would have a model that is 29,035 feet high. I need to use a smaller unit of measurement,” Josh said.
“What about @$\begin{align*}\frac{1}{4}\end{align*}@$?” Sasha suggested.
“That’s possible, but it still will probably be too big,” he stopped talking to scribble some numbers down in his notebook. “I think that @$\begin{align*}\frac{1}{8}^{\prime\prime} = 2000 \ feet\end{align*}@$ will be perfect.”
Sasha looked puzzled.
Do you understand what Josh was working on? How did he come to that scale? Why do you think Sasha is puzzled? What will the actual dimensions of the model be? To answer these questions, you will need to understand scale, unit scale and ratios. Pay attention to the information presented in this Concept and you will be all ready to answer these questions by the end.
Guidance
A ratio is a comparison between two quantities. We can write a ratio in fraction form, by using a colon or by using the word “to”.
Sometimes in life, we have a real-life object that we want to represent in a smaller form. Think about buildings. We can’t build an actual building to show the dimensions in a smaller way, so we build a model of the building. When we do this, we take the actual dimensions and shrink them down to build a model. When we do this, we create a unit scale for the model. When we create a unit scale, we decide on a measurement to represent an actual measurement.
That’s a great question. First, let’s look at a unit scale.
1 inch = 3 feet
This is a unit scale. We have a unit represented by the one inch. Remember that when we talk about unit, we are talking about a relationship to one. We have one inch represented by three feet.
The one inch is the scale dimension and the three feet is the actual dimension we are measuring.
Now, not all objects that you will create a model of will measure exactly what the unit scale does, so we have to use a unit scale to show the relationship between scale dimensions and actual dimensions. Scale dimensions are the dimensions of the model, and actual dimensions are the real – life dimensions.
Using the unit scale above, what would be the relationship between the scale dimensions and the actual dimensions for an object 24 feet long?
First, let’s think about our unit scale.
1 inch = 3 feet
If we have a building 24 feet long, that is the actual dimensions. We need to represent the dimensions using our unit scale.
We can say that 8 inches = 24 feet. This is our answer.
If we know the scale dimensions and the unit scale, then we can find the actual dimensions.
It’s a little like figuring out a puzzle. We have to have the pieces of the puzzle to put the puzzle together. What are the pieces that we need?
Necessary Information
- To find the scale dimensions, we need the unit scale and the actual dimensions
- To find the actual dimensions, we need the unit scale and the scale dimensions.
Write this necessary information down in your notebooks.
What is the scale length of the object if the unit scale is 2 inches : 4 feet and the actual dimensions of the object is 20 feet?
First, let’s make sure that we have been given all of the necessary information. First, the unit scale has been given.
2 inches : 4 feet
So for every four feet of the building, we are going to have 2 inches of our model.
The actual length of the building has also been given. It is 20 feet. We can figure this out using a proportion.
@$$\begin{align*}\frac{scale \ dimension}{actual \ dimension} = \frac{2 \ in}{4 \ ft} = \frac{x}{20 \ ft}\end{align*}@$$
Notice that we used a different form of the ratio to solve this proportion. Now we can solve it. Four times five equals twenty, so we can do this to the top measurement. Two times five is equal to 10.
The scale dimension for length would be 10 inches.
Find the scale dimension for each situation if the scale is 1" = 5 feet.
Example A
25 feet
Solution: 5 inches
Example B
3 inches
Solution: 15 feet
Example C
75 feet
Solution: 15 inches or 1 foot, 3 inches
Now let's go back to the dilemma from the beginning of the Concept.
First, let’s write a ratio to show the scale that Josh has selected.
@$\begin{align*}\frac{1}{8}^{\prime\prime} = 2000 \ feet\end{align*}@$
@$$\begin{align*}\frac{\frac{1}{8}^{\prime\prime}}{2000 \ ft}\end{align*}@$$
Next, we write a ratio to show how many feet are in @$\begin{align*}1^{\prime\prime}\end{align*}@$. We can use a proportion to complete this task.
@$$\begin{align*}\frac{\frac{1}{8}^{\prime\prime}}{2000 \ ft} = \frac{1^{\prime\prime}}{x}\end{align*}@$$
If we cross multiply and divide, we can see that @$\begin{align*}1^{\prime\prime} = 16,000 \ feet\end{align*}@$.
Now we can use this information to figure out the dimensions of the model. Once again, use a proportion.
@$$\begin{align*}\frac{1^{\prime\prime}}{12,000 \ ft} = \frac{x}{29,035 \ ft}\end{align*}@$$
We cross multiply and divide.
@$$\begin{align*}12,000x &= 29,035\\ x &= 1.8 \ feet\end{align*}@$$
Josh’s model will be 1.8 feet tall, a workable size for a model.
Guided Practice
Here is one for you to try on your own.
Using a unit scale of 1 inch : 8 feet, what is the actual dimension of an object with a scale dimension for length of 5 inches?
Solution
First, let’s make a note of the unit scale.
1 inch = 8 feet
We have been given the scale dimension and not the actual dimension. We are going to need to solve for the actual dimension. Let’s write a proportion to do this.
@$$\begin{align*}\frac{scale \ dimension}{actual \ dimension} = \frac{1 \ in}{8 \ ft} = \frac{5 \ in}{x}\end{align*}@$$
We can see that one times five is equal to five. We can do this to the actual dimension as well. Eight times five is equal to forty.
The actual length of the building is 40 feet.
Video Review
Explore More
Directions: Find the scale dimension given the scale. Write a proportion and an answer for each problem. There are two answers for each problem.
- Scale is @$\begin{align*}1^{\prime\prime} = 2 \ ft\end{align*}@$, the actual dimension is 18 feet
- Scale is @$\begin{align*}1^{\prime\prime} = 5 \ feet\end{align*}@$, the actual dimension is 20 feet
- Scale is @$\begin{align*}\frac{1}{2}^{\prime\prime} = 2 \ feet\end{align*}@$, actual dimension is 10 feet
- Scale is @$\begin{align*}1^{\prime\prime} = 12 \ feet\end{align*}@$, actual dimension is 72 feet
- Scale is @$\begin{align*}3^{\prime\prime} = 4 \ feet\end{align*}@$, actual dimension is 16 feet
Directions: Using a scale of 1 to 2, figure out the actual dimensions given each scale.
- 4 to _____
- 6 to _____
- 9 to _____
- 12 to _____
- 14 to _____
Directions: Using a scale of 3 to 4, figure out the actual dimensions given each scale.
- 6 to _____
- 9 to _____
- 12 to _____
- 18 to _____
- 36 to _____