Plans and Scale

A group of archaeologists has drawn an ancient vase on paper. In the drawing, the vase is $15\text{ cm}$ high. If we know that the actual vase has a height of $75\text{ cm}$, what scale have the archaeologists used?

In a model of a famous communications tower, the height of the replica is $85$ cm. If we know that the actual height of the tower is $255$ meters, determine the simplified numerical scale to which the model was built.

On a road map of Mexico, the given scale is $1:250,000$. If you use a ruler to measure the distance between two towns and measure exactly $8\text{ cm}$, how many kilometers apart are these two towns in reality?

On a hiking map of the Nevado de Toluca, the scale indicated is $1:25,000$. If the distance measured with a ruler between the Parque de los Venados and the Laguna del Sol is $12.4$ cm on paper, how many actual kilometers will a group of hikers have to travel to go from one point to the other in a straight line?

On a road map of Mexico with a scale of $1:2,500,000$, the distance measured with a ruler between Mexico City and Guadalajara is $14.2\text{ cm}$. A delivery truck leaves Mexico City at $08:00\text{ h}$ and travels toward Guadalajara at an average speed of $90\text{ km/h}$.

What time is it expected to arrive at its destination if it stops for half an hour halfway through to rest?

A biologist photographs a small queen ant using an enlargement scale of $5:1$. If in the photograph the ant measures $45\text{ mm}$ in length, what is its actual size in millimeters?

A model car collector has a scale model of $1:24$. If the length of the miniature is $18\text{ cm}$, how long will the actual car be in meters?

A biologist observes an ant through a microscope. In their technical drawing, the ant has a length of $12$ cm. If the scale of the drawing is an enlargement of $25:1$, what is the actual length of the ant expressed in millimeters?

You want to make a drawing in your notebook of your new graphics tablet. The tablet has an actual length of $30\text{ cm}$. If you decide to use a reduction scale of $1:3$, what will the length of the drawing in your notebook be?

On the floor plan of a house drawn to a scale of $1:100$, the main bedroom is represented as a rectangle measuring $4\text{ cm}$ long and $3\text{ cm}$ wide. Calculate the actual dimensions of the room expressed in meters.

You have decided to redecorate your room and want to draw a scale plan at $1:20$ to see if a new desk fits. If your room is rectangular and measures $4.2$ meters long by $3.4$ meters wide, what will the dimensions (length and width) of the rectangle in your drawing be, expressed in centimeters?

You see a linear scale on a hiking map where a segment of $1\text{ cm}$ is equivalent to $500\text{ meters}$ in reality. If the route you want to do measures $12\text{ cm}$ on the map, how many kilometers will you walk in total?

A road map has a linear scale in which a segment of $2$ cm represents an actual distance of $15$ km. If the distance between two cities on that map is $9.4$ cm, what is the actual distance in kilometers between both cities?

An old map has lost its numerical scale indication. However, it shows a linear scale in which a line segment measuring $3.5\text{ cm}$ on paper represents an actual distance of $17.5\text{ kilometers}$.

1. Determine the numerical scale of the map.

2. If two towns are separated on the map by $12\text{ cm}$, how many hours will it take a cyclist to go from one to the other at a constant speed of $24\text{ km/h}$?

You have two different maps of the same area. On Map A, the distance between two villages is $6$ cm and the scale is $1:50,000$. On Map B, the distance between the same villages is $15$ cm. Calculate the numerical scale of Map B.

A forest monitoring drone flies along a rectangular route over a nature park. At the control center, technicians view the route on a digital screen where the scale is $1:50,000$. On this screen, the rectangle measures $12\text{ cm}$ long and $8\text{ cm}$ wide. However, for the printed report, they must reduce the map so that the length of the route is only $4\text{ cm}$ long.

Calculate:

1. The actual area of the nature park in square kilometers ($\text{km}^2$).

2. The numerical scale that the map in the printed report should have.

A photocopy has been made of an original plan that was at a scale of $1:200$. Due to a setting error, the photocopy has come out reduced to $75\%$ of its original size.

On the photocopy, the living room of the house measures $6\text{ cm}$ in length.

1. What is the actual scale of the photocopy compared to reality?

2. What is the actual length of the living room in meters?

The plan of a local soccer field is drawn to a scale of $1:400$. On the plan, the area occupied by the grass is a rectangle $25$ cm long by $15$ cm wide. Calculate the actual area of the soccer field in square meters.

You are designing your own "Gaming Room" on a scale plan at $1:40$. In the drawing, the room has an area of $75\text{ cm}^2$. You have seen a professional desk that takes up an actual area of $1.28\text{ m}^2$ and a chair that takes up $0.6\text{ m}^2$.

If you decide to place the desk and two identical chairs, what actual area will remain free in the room? Express your answer in square meters ($\text{m}^2$).

An architect has built a scale model of a new auditorium at a scale of $1:250$. The total volume of the model is $0.64\text{ dm}^3$.

Given that the density of the actual materials is different, but the volume remains proportional to the cube of the scale factor, calculate the actual volume of the auditorium in cubic meters ($\text{m}^3$).

(Note: Remember that if the length scale factor is $k$, the volume scale factor is $k^3$).