Plans and scale

On a road map of Ireland, 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 kilometres apart are these two towns in reality?

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 $255$ metres, determine the simplified numerical scale to which the model was built.

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 metres.

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 metres?

On a hiking map of the Wicklow Mountains, the scale indicated is $1:25,000$. If the distance measured with a ruler between Glendalough and Lugnaquilla is $12.4$ cm on paper, how many actual kilometres will a group of hikers have to travel to go from one point to the other in a straight line?

Cian is designing the floor plan of his bedroom at a scale of $1:50$ in his Technology class. Measuring the main wall where he will place the bed with a ruler, he records a length of $7.2\text{ cm}$. Next, he performs the following calculation to find the real-life measurement: $7.2 \times 50 = 360$, and concludes in his notebook: "The wall of my room is $360\text{ meters}$ long".

1. Explain with reasoning why the result Cian reaches lacks physical sense for a home.

2. Identify the specific unit or conceptual error he has made when interpreting the scale.

3. Correct his procedure and determine the correct real length of the wall expressed in meters.

You are planning a hiking route through the Cabo de Gata Natural Park (Almería) using a topographic map at a scale of $1:40,000$. With a ruler, you measure a straight-line distance of $7.5\text{ cm}$ between two coastal coves.

1. Estimate: Without using a calculator or performing complex multiplications with decimals, make a preliminary mental estimate. Knowing that at a scale of $1:40,000$ each centimeter is equivalent to just under half a kilometer ($0.4\text{ km}$), will the real distance be closer to $1\text{ km}$, $3\text{ km}$, or $10\text{ km}$? Explain your reasoning.

2. Calculate: Apply scale relationships to find the exact value of the real distance in kilometers.

3. Check: Compare your calculation with your initial estimate and argue whether a person walking at a steady pace of $4\text{ km/h}$ would take more or less than one hour to walk that route in a straight line.

A young architect is designing a new skate park for her city. On the technical plan, the main jump track has a length of $15$ cm. If the park is going to be built to a scale of $1:150$, calculate the actual length of the track in metres.

You see a linear scale on a hiking map where a segment of $1\text{ cm}$ is equivalent to $500\text{ metres}$ in reality. If the route you want to do measures $12\text{ cm}$ on the map, how many kilometres 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 kilometres between both cities?

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?

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 millimetres?

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 millimetres?

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 centre, 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 kilometres ($\text{km}^2$).

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 metres?

The Sports department of a town council is going to publish an informative brochure containing the plan of a standard outdoor multi-sports court. The real dimensions of the court are $40\text{ m}$ long by $20\text{ m}$ wide. In the print layout, they have reserved a closed box measuring $12\text{ cm}$ long by $8\text{ cm}$ wide to insert this plan. The designer can only choose among four standard normalized scales: $1:200$, $1:300$, $1:400$, and $1:500$.

1. Analyze each scale and determine which is the largest scale (the one with the highest level of detail) that allows the entire court to be drawn inside the box without exceeding any of its margins.

2. Using the appropriate scale you selected, calculate the exact dimensions (length and width) that the drawn rectangle will have on paper in centimeters.

3. How many centimeters of free margin will remain along the length and the width inside the brochure's box?

On a road map of Ireland with a scale of $1:2,500,000$, the distance measured with a ruler between Dublin and Cork is $14.2\text{ cm}$. A delivery lorry leaves Dublin at $08:00\text{ h}$ and travels towards Cork 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?

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 $17.5\text{ kilometres}$.

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 $24\text{ km/h}$?

The plan of a local football pitch 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 football pitch in square metres.

Through the online portal of Tailte Éireann, an agricultural cooperative in County Kilkenny consults a rectangular barley field represented on a map at a scale of $1:5,000$. On the screen, the rectangle representing the estate measures $8\text{ cm}$ long and $5\text{ cm}$ wide.

1. Model the situation mathematically and calculate the real dimensions of the estate (length and width) expressed in meters.

2. Obtain the total real surface area of the plot expressed in square meters ($\text{m}^2$) and convert it to hectares ($1\text{ ha} = 10,000\text{ m}^2$).

3. Teagasc reports for the area indicate an average yield of $4,200\text{ kg}$ of barley per hectare, and it is known that the maltings extracts $20\,\%$ of its weight as malt for distilling. Calculate how many kilograms of malt the cooperative expects to produce from the harvest of that plot.

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 metres ($\text{m}^2$).

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 metres ($\text{m}^3$).

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