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It is sometimes utterly impossible to accurately model an object or real-life scenario using the original dimensions. Here, scale drawings and models come into action. This lesson will expand on how these tools are instrumental in modeling some common cases.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Challenge

On a sunny Saturday morning, Kevin — a boy with a passion for formula $1$ racing cars — wakes up and finds a letter sitting on the table. It is from his pops.

The first task Kevin faces is to get to his uncle's bakery without the help of his phone's GPS. He is given two things: a tape measure and a map with a marked route.
When Kevin arrives at the bakery, his uncle asks him how far he walked from home. If Kevin answers correctly, he will get the next task in the scavenger hunt!

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Discussion

Most maps include the math of a particular relation between two units of measure. One of the measures refers to distances on the map itself and the other refers to actual distances. This relationship is called a *scale.* Other real-life tools use this relationship as well.

Concept

The scale of a model or drawing is the ratio between any length on the model or drawing and its corresponding length on the actual object or place.

$Length on thedrawing :Corresponding lengthon the actual object $

Suppose a drawing has a scale of $1in:100ft.$ This means that $1$ inch on the drawing represents $100$ feet on the actual object. Apart from the colon notation, a scale can be expressed using an equals sign or as a fraction, as it is a ratio.

Denoting a Scale | |
---|---|

Ratio | $1in:100ft$ |

Equals Sign | $1in=100ft$ |

Fraction | $100ft1in $ |

When a scale is written without specifying the units, it is understood that both numbers have the same units of measure. For example, a scale of $1:2$ means that the actual object is twice the size of the model. A scale of $1:0.5$ means that the actual object is half the size of the model.

External credits: Derek Quinn

Discussion

A scale drawing is a two-dimensional drawing that is similar to an actual object or place. In a scale drawing, the ratio of any length on the drawing to the actual length is always the same and equal to the scale of the drawing.

$actual→drawing→ L_{1}ℓ_{1} =L_{2}ℓ_{2} ←actual←drawing $

Possible examples of scale drawing are floor plans, blueprints, and maps. $90yd1.5in =ℓ15.65in $

By solving this equation for $ℓ,$ the distance Kevin walked from his house to his uncle's bakery can be found.
$90yd1.5in =ℓ15.65in $

$ℓ=939yd$

Example

After correctly determining the distance traveled to the bakery, Kevin's uncle trades him the tape measure for a smartwatch and a new map. Kevin's next task is to reach his aunt's house. There is one catch — his route has to pass through the local bank.
*on the map* in inches? Kevin has to answer correctly to continue the scavenger hunt. ### Hint

### Solution

Since a map is a scale drawing, the ratio of any length on the drawing to the actual length always remains the same. Based on that, the following equation is created.
On the map, the bank and the aunt's house are $2.66$ inches apart.

Kevin makes it to his aunt's house! She asks him how is her house from the bank

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A map is a scale drawing. Then, the ratio of any length on the drawing to the actual length is always the same. The watch shows the distances traveled, where the first distance indicates the distance between the bakery and the bank.

Kevin's smartwatch gives the actual distances between the visited places. According to the map, the bakery and the bank are $2.85$ inches apart. Let $x$ be the distance between the bank and the aunt's house on the map.

Place $1$ | Place $2$ | Distance on the Map (in) | Actual Distance (mi) |
---|---|---|---|

Bakery | Bank | $2.85$ | $0.75$ |

Bank | Aunt's house | $x$ | $0.70$ |

$actual→drawing→ 0.752.85 =0.70x ←actual←drawing $

The value of $x$ can be found by solving this equation.
$0.752.85 =0.70x $

▼

Solve for $x$

MultEqn

$LHS⋅0.70=RHS⋅0.70$

$0.752.85 ⋅0.70=x$

CalcQuot

Calculate quotient

$3.8⋅0.70=x$

Multiply

Multiply

$2.66=x$

RearrangeEqn

Rearrange equation

$x=2.66$

Discussion

If the original real-life situation involves a three-dimensional object, making a *scale model* is more useful than a drawing. The idea behind a scale model is the same as a scale drawing, but the model has three dimensions instead of two.

Concept

A scale model is a three-dimensional model that is similar to a three-dimensional object. The ratio of a linear measurement of a model to the corresponding linear measurement of the actual object is always the same and equal to the scale of the model.

$actual→model→ L_{1}ℓ_{1} =L_{2}ℓ_{2} ←actual←model $

Here is an example scale model of a building. Example

The lighthouse door is closed and the code to open it is the actual height of the lighthouse, measured in meters. For a moment, Kevin does not know what to do. He realizes that he should open the backpack his aunt gave him. Inside, he finds a scale model of the lighthouse and a tape measure.

External credits: Ekayasa.Design

Kevin figures it out and manages to measure the width of the base of the lighthouse. Also, he measures the width and height of the scale model with the tape measure.

Scale Model | Lighthouse | |
---|---|---|

Width | $10cm$ | $5.25m$ |

Height | $30cm$ | $?$ |

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In a scale model, the ratio of a linear measurement on the model to the corresponding linear measurement on the actual object is always the same.

Let $x$ be the height of the lighthouse. In a scale model, the ratio of a linear measurement on the model to the corresponding linear measurement on the actual object is always the same. This leads to writing the following equation.
The height of the lighthouse is $15.75$ meters. Kevin is ready to enter the lighthouse.

$actual→model→ 5.25m10cm =x30cm ←actual←model $

The height of the lighthouse can be found by solving the previous equation for $x.$
$5.2510 =x30 $

CrossMult

Cross multiply

$10x=5.25⋅30$

Multiply

Multiply

$10x=157.5$

DivEqn

$LHS/10=RHS/10$

$x=10157.5 $

CalcQuot

Calculate quotient

$x=15.75$

Discussion

The length scale factor of a scale drawing or scale model is the ratio of a length on the drawing or model to the corresponding actual length where both lengths have the *same* units of measure.

$Length scale factor=Actual lengthLength on model $

Example

At the top of the lighthouse, Kevin finds a blueprint of his room and a scale model of his house shed. He understands that he must return home.
Kevin uses the tape measure and finds that his bed is $1.80$ meters long. He then goes to the backyard and measures the width of the shed. It is $4.50$ meters wide. What is the pin of the padlock? ### Hint

### Solution

Before finding the length scale factor, all the dimensions should be written with the same units of measure. Use the fact that $1$ meter is the same as $100$ centimeters to convert $1.80$ meters to centimeters.

As before, write all the dimensions using the same units of measure.
The pin to open the padlock is $13.$

In his room, Kevin finds a chest closed with a padlock. The pin to open it is $156$ times the sum of the length scale factors of the blueprint and the scale shed.

External credits: juicy_fish

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Be sure all the dimensions are written with the same units of measure before finding the length scale factor.

The length scale factor of the blueprint is the ratio of a dimension on the blueprint to the corresponding actual dimension. The dimensions of the bed on the blueprint are written on its side. Additionally, Kevin measured his actual bed length.

Blueprint Dimensions (cm) | Actual Dimensions (m) | |
---|---|---|

Bed length | $9$ | $1.80$ |

Bed width | $5$ |

$1.80m⋅1m100cm =180cm $

Next, divide the bed length on the blueprint by the actual length of the bed to find the scale factor.
The length scale factor of the blueprint is $201 .$ This means that the actual dimensions of the bed — and the entire room — were divided by $20$ to make the blueprint.
Similarly, the length scale factor of the scale shed can be found. The width and length of the model are known and Kevin measured the actual width of the shed.

Scale Model Dimensions (cm) | Actual Dimensions (m) | |
---|---|---|

Length | $20$ | |

Width | $15$ | $4.50$ |

$4.50m⋅1m100cm =450cm $

The length scale factor is the quotient between the width of the scale shed and the actual width of the shed.
Now that both length scale factors are known, the pin of the padlock can be found. Recall, it is $156$ times the sum of the length scale factors.
$Pin=156(scale factor1+scale factor2)$

SubstituteValues

Substitute values

$Pin=156(201 +301 )$

▼

Simplify

ExpandFrac

$ba =b⋅30a⋅30 $

$Pin=156(20⋅301⋅30 +301 )$

ExpandFrac

$ba =b⋅20a⋅20 $

$Pin=156(20⋅301⋅30 +30⋅201⋅20 )$

Multiply

Multiply

$Pin=156(60030 +60020 )$

AddFrac

Add fractions

$Pin=156⋅60050 $

ReduceFrac

$ba =b/50a/50 $

$Pin=156⋅121 $

MoveLeftFacToNumOne

$a⋅b1 =ba $

$Pin=12156 $

CalcQuot

Calculate quotient

$Pin=13$

Example

Kevin eagerly opens the chest. He discovers a tablet! Upon powering it on, a map with a marker at his current position and another marker at his next stop appear. Kevin then presses the navigate button. Oh no! The app locks and asks him what the actual distance is between the marked places, in yards.
According to the information given by the app, with the current zoom, the markers are $9$ inches apart on the screen and the length scale factor is $0.0004.$ What pin unlocks the app? ### Hint

### Solution

External credits: Freepik

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The ratio of the distance on the tablet to the actual distance is equal to the length scale factor. Use the fact that $36$ inches are the same as $1$ yard.

Since a map is a scale drawing, the ratio of any length on the map to the actual length is always the same and equal to the length scale factor.
The places are $22500$ inches apart. This can be converted to yards by using the fact that $36$ inches are the same as $1$ yard.

$Corresponding actual lengthLength in map =k $

This means that the ratio of the distance between the marked places on the tablet to the actual distance is equal to the length scale factor. Let $x$ be the actual distance. According to the information given by the app, the places are $9$ inches apart on the screen and the scale factor is $0.0004.$
$Actual lengthLength in map =k$

SubstituteValues

Substitute values

$x9in =0.0004$

▼

Solve for $x$

MultEqn

$LHS⋅x=RHS⋅x$

$9in=0.0004⋅x$

DivEqn

$LHS/0.0004=RHS/0.0004$

$0.00049in =x$

CalcQuot

Calculate quotient

$22500in=x$

RearrangeEqn

Rearrange equation

$x=22500in$

$22500in⋅36in1yd =625yd $

The actual distance between the marked places is $625$ yards. This means that the pin that unlocks the app is $625.$ Enter it to see the route and follow Kevin's adventure.
Example

When Kevin arrived at the last destination, his father was there waiting with a huge smile and a box in his hands. Inside, there was a scale formula $1$ racing car made with a length scale factor of $161 !$

External credits: kjpargeter

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Let $x$ be the length of the scale car, in centimeters. Since the length of the actual car is given in meters, convert it to centimeters first.
The length of the scale formula $1$ is $35$ centimeters.

$5.6m⋅1m100cm =560cm $

The actual formula $1$ car is $560$ centimeters long. The ratio of the length of the scale car to the length of the actual car is equal to the length scale factor, $161 .$
$Length Scale factor=560x $

Substitute

$Length Scale factor=161 $

$161 =560x $

▼

Solve for $x$

MultEqn

$LHS⋅560=RHS⋅560$

$161 ⋅560=x$

MoveRightFacToNumOne

$b1 ⋅a=ba $

$16560 =x$

CalcQuot

Calculate quotient

$35=x$

RearrangeEqn

Rearrange equation

$x=35$

Closure

The length scale factor gives the relationship between the dimensions of a scale drawing and the original drawing. Kevin wonders whether the areas are also related somehow. Use the following applet to investigate it.

Notice that the ratio of the scale triangle's area to the original area is equal to the square of the length scale factor. This relationship holds for any scale drawing. Finally, think about whether there is also a relationship between a scale model's volume and the actual object's volume.

In the following table, some ratios of the area of the scale triangle to the area of the original triangle are computed.

Area of Original Triangle | Length Scale Factor | Area of Scale Triangle | Ratio of Areas |
---|---|---|---|

$2.8$ | $2$ | $11.2$ | $2.811.2 =4=2_{2}$ |

$2.8$ | $0.5=21 $ | $0.7$ | $2.80.7 =0.25=41 =(21 )_{2}$ |

$2.8$ | $1.5=23 $ | $6.3$ | $2.86.3 =2.25=49 =(23 )_{2}$ |

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