Exercises: Plans and scale

On a road map of the US, the indicated scale is $1:250.000$. If you use a ruler to measure the distance between two towns and mark exactly $8\text{ in}$, how many miles separate these two towns in reality?

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

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

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

A miniature car collector has a model at a scale of $1:24$. If the length of the miniature is $18\text{ in}$, how long will the real car measure in feet?

You observe a graphic scale on a hiking map where a $1\text{ in}$ line segment is equivalent to $500\text{ meters}$ in reality. If the route you want to do measures $12\text{ in}$ on the map, how many miles 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 $45\text{ in}$ in length, what is its real size in inches?

In a community center brochure, they have drawn a basketball court at a scale of $1:200$. If the real court measures $28\text{ meters}$ long, how many inches will the length of the court measure in the brochure's drawing?

On a hiking map of the Rocky Mountains, the indicated scale is $1:25.000$. If the distance measured with a ruler between Bear Mountain and Eagle Peak is $12,4$ in on the paper, how many real miles 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 $15$ in. If the park is going to be built at a scale of $1:150$, calculate the real length of the track in feet.

You have decided to change the decoration of your room and you want to draw a plan at a scale of $1:20$ to see if a new desk fits. If your room is rectangular and measures $4,2$ feet long by $3,4$ feet wide, what will the dimensions (length and width) of the rectangle in your drawing be, expressed in inches?

In the model of a famous communications tower, the height of the replica is $85$ in. If we know that the real height of the tower is $255$ feet, determine the simplified numerical scale at which the model has been built.

A road map has a graphic scale in which a $2$ in line segment represents a real distance of $15$ mi. If the distance between two cities on that map is $9,4$ in, what is the real distance in miles between both cities?

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

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

You have two different maps of the same area. On Map A, the distance between two towns is $6$ in and the scale is $1:50.000$. On Map B, the distance between those same towns is $15$ in. Calculate what the numerical scale of Map B is.

A forest surveillance drone flies a rectangular route over a nature reserve. In the control center, technicians view the route on a digital screen where the scale is $1:50.000$. On this screen, the rectangle measures $12\text{ in}$ long and $8\text{ in}$ wide. However, for the printed report, they must reduce the map so that the length of the route measures only $4\text{ in}$.

Calculate:

1. The real area of the nature reserve in square miles ($\text{mi}^2$).

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

You are designing your own "Gaming Room" on a plan with a scale of $1:40$. In the drawing, the room has an area of $75\text{ in}^2$. You have seen a professional desk that occupies a real area of $1,28\text{ ft}^2$ and a chair that occupies $0,6\text{ ft}^2$.

If you decide to place the desk and two identical chairs, what real area will be left free in the room? Express the result in square feet ($\text{ft}^2$).

On a road map of the US with a scale of $1:2.500.000$, the distance measured with a ruler between Dallas and Houston is $14,2\text{ in}$. A delivery truck leaves Dallas at $08:00\text{ h}$ and travels towards Houston at an average speed of $90\text{ mph}$.

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

A map app 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 subway stations is $6\text{ in}$. If the user zooms out until the scale is $1:50.000$ (level B):

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

2. If the phone screen has an area of $80\text{ in}^2$, how many real square miles of urban area are being viewed at level B?

An architect has built a model of a new auditorium at a scale of $1:250$. The total volume of the model is $0,64\text{ in}^3$.

Knowing that the density of the real materials is different, but the volume remains proportional to the cube of the scale, calculate the real volume of the auditorium in cubic feet ($\text{ft}^3$).

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

The plan of a municipal swimming pool is drawn to a scale such that one of its walls, which measures $25\text{ meters}$ in reality, is represented by a $12,5\text{ cm}$ line segment. On the plan, the bottom of the pool has an area of $120\text{ cm}^2$.

They want to cover the real bottom with square tiles of side $20\text{ cm}$. How many tiles will they need to buy?

An old map has lost its numerical scale indication. However, it shows a graphic scale in which a line segment measuring $3,5\text{ in}$ on the paper represents a real distance of $17,5\text{ kilometers}$.

1. Determine the numerical scale of the map.

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

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

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

1. What is the real scale of the photocopy with respect to reality?

2. What is the real length of the living room in feet?