Exercises: 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?

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 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?

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?

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 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?

In a sports centre brochure, a basketball court has been drawn to a scale of $1:200$. If the actual court is $28\text{ metres}$ long, how many centimetres long will the court be in the drawing in the brochure?

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.

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?

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 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 $4.2$ metres long by $3.4$ metres wide, what will the dimensions (length and width) of the rectangle in your drawing be, expressed in centimetres?

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.

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?

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?

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.

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.

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?

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.

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.

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

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?

A map application on a smartphone shows an area of the city. When the zoom is at level A, the scale is $1:15,000$ and the distance between two metro stations is $6\text{ cm}$. If the user zooms out until the scale is $1:50,000$ (level B):

1. What will the distance in centimetres between those same stations be on the screen?

2. If the phone screen has an area of $80\text{ cm}^2$, how many actual square kilometres of urban surface are being displayed at level B?

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 $k^3$).

The plan of a municipal swimming pool is drawn to such a scale that one of its walls, which measures $25\text{ metres}$ in reality, is represented by a line segment of $12.5\text{ cm}$. On the plan, the bottom of the pool has an area of $120\text{ cm}^2$.

They want to cover the actual bottom with square tiles measuring $20\text{ cm}$ on each side. How many tiles will need to be bought?

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 $24\text{ km/h}$?

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 urban planning department has the plan of a future pedestrian square at a scale of $1:200$. On this plan, the square covers an area of $64\text{ cm}^2$. For a neighborhood exhibition, they need to draw an enlargement of the same square on a panel at a scale of $1:50$. Two draftsmen argue about which method is the most convenient to find the area the square will occupy on the new panel:

• Method A (Passing through reality): Calculate the real surface area of the square in $\text{m}^2$ from the first plan and its scale ($1:200$) and, from that real data, deduce the surface area it will occupy on the panel at a scale of $1:50$.

• Method B (Direct ratio of areas): Directly compare the two scales by noting that $200 \div 50 = 4$ (lengths on the panel are quadrupled) and directly apply the ratio between areas by multiplying the initial area by $4^2 = 16$.

• Solve the problem by applying Method A step by step, stating the real surface area in $\text{m}^2$ and the final surface area on the panel in $\text{cm}^2$.

• Solve the problem by applying Method B and check whether both results match exactly.

• Compare both procedures: which one reduces the risk of making unit conversion errors, and which one would be essential if we were also asked to budget the cost of paving the real square?

During a high-temperature event, Civil Protection monitors an active wildfire using an emergency application configured at a scale of $1:25,000$. On the tactical map, the burned area is approximated by a trapezoid with bases of $6\text{ cm}$ and $10\text{ cm}$, and a height of $4\text{ cm}$.

1. Calculate the area occupied by the fire on the map in $\text{cm}^2$ and, using the quadratic scale relationship ($\text{Real area} = \text{Area on plan} \times k^2$), calculate the real burned surface area expressed directly in hectares ($1\text{ ha} = 10,000\text{ m}^2$). Justify the conversion factor used.

2. On the same tactical map, the advancing front that ground crews must tackle measures a segment of $12\text{ cm}$. Incident commanders request a printed pocket sheet for the crews on a laminated card whose maximum available usable space is $15\text{ cm}$. If a standard technical scale $1:k$ must be used where $k$ is an exact multiple of $5,000$ ($1:15,000$, $1:20,000$, $1:25,000$, $1:30,000$...), determine the scale with maximum detail (lowest value of $k$) that allows this entire front to be represented without exceeding $15\text{ cm}$, and calculate the exact length this front will have on the sheet.

Unit Challenge. Design your bedroom renovation!

1. Floor plan and scale: Measure the real dimensions of your room with a tape measure. Draw its floor plan at a scale of $1:50$ (by hand or with GeoGebra) and calculate the ratio between the real surface area and that of the drawing.

2. Budget and yield: Look up the price of a can of paint on a retail website. Knowing that $2$ cans cover $15.5\text{ m}^2$ by applying $2$ coats in $4$ hours of work, calculate using a compound rule of three how many cans and hours you would need to paint your room's ceiling (same area as the floor plan) with $3$ coats. Apply a $10\%$ discount to the final cost and add $21\%$ VAT.

3. Distribution and modeling: In a spreadsheet, tabulate and plot the inverse proportionality function $t = k/n$ (time spent according to the number $n$ of people painting). Finally, simulate financing the total expense through a directly proportional division between two relatives according to their ages.

Present a short report with the floor plan, the table with the graph, and the detailed calculations.

Assessment criteria:

• Precision in calculating the scale, the ratio, and drawing the floor plan.

• Correctness and rigor in the compound rule of three and the handling of percentages.

• Appropriate representation of the inverse function and resolution of the proportional distribution.

• Clarity of explanation and presentation of data in the report.