Exercises: Length

In the study of math and physics, it is essential to distinguish between what we measure and the unit we use. Classify the following concepts in your notebook according to whether they are a quantity or a unit of measurement:

• Length

• Meter ($m$)

• Centimeter ($cm$)

• Width of a table

• Kilometer ($km$)

• Distance between two cities

To measure accurately, we must choose the most appropriate unit. State which unit of the metric system ($mm$, $cm$, $m$ or $km$) you would use to measure the following objects:

1. a) The length of a paperclip.
2. b) The height of your best friend.
3. c) The distance a school bus travels between two towns.
4. d) The thickness of the screen of a latest-generation cell phone.

The metric system is decimal because each unit is $10$ times greater than the previous one. Complete the following equalities by converting to the indicated unit:

1. a) $7\text{ m} = \dots\dots\dots\text{ dm}$
2. b) $25\text{ cm} = \dots\dots\dots\text{ mm}$
3. c) $1,5\text{ m} = \dots\dots\dots\text{ cm}$
4. d) $0,8\text{ dm} = \dots\dots\dots\text{ mm}$

When we measure large distances, we use multiples of the meter. Perform the following conversions:

1. a) $3.000\text{ m} = \dots\dots\dots\text{ km}$
2. b) $450\text{ m} = \dots\dots\dots\text{ hm}$
3. c) $12\text{ dam} = \dots\dots\dots\text{ m}$
4. d) $0,5\text{ km} = \dots\dots\dots\text{ hm}$

Three friends compare the length of their earphone cables to see which is longest. The measurements are:

• Marcus: $1,2\text{ m}$

• Lucy: $125\text{ cm}$

• Hugo: $11,5\text{ dm}$

Express all measurements in centimeters and order them from shortest to longest using the $<$ symbol.

A stage of a hiking route through the Blue Ridge Mountains is divided into two sections. The first section measures $3\text{ km y } 450\text{ m}$, and the second section measures $2.100\text{ m}$. How many kilometers does the complete stage measure?

Sarah is designing a case for her tablet. She uses a ruler and observes that the length of the tablet is $24,5\text{ cm}$. If she wants to put a protective edge of $5\text{ mm}$ on each of the two ends of the length, what will be the total length of the case in centimeters?

Calculate the perimeter (the sum of all its sides) of a futsal court whose measurements are $40\text{ m}$ long and $200\text{ dm}$ wide. Express the final result in meters.

(Hint: Remember that to add measurements they must all be in the same unit).

Three friends are comparing the distances they ran this morning while training for a charity race. Ashley says she ran $4,2 \text{ km}$, Marcus claims to have done $38 \text{ hm}$ and $15 \text{ dam}$, and Sophia recorded a total of $4.050 \text{ m}$ on her smartwatch.

Order the three friends from greatest to shortest distance run, expressing all measurements in meters so they can be compared.

A delivery drone makes a trip divided into three stages. In the first stage, it travels a distance of $1 \text{ km}$ and $25 \text{ m}$. In the second, it flies $85 \text{ hm}$ and $7 \text{ dam}$. In the last stage, it travels another $450 \text{ m}$.

Calculate the total distance of the trip, expressing the result in simple form (a single unit) in kilometers.

In an electronics store, monitor screens are measured by the length of their diagonal in inches. We know that $1 \text{ pulgada} = 2,54 \text{ in}$.

If we buy a monitor whose diagonal measures $27 \text{ pulgadas}$, what is its approximate measurement in millimeters?

A carpenter has a wooden board $3 \text{ m}$ long. He needs to cut pieces of $35 \text{ cm}$ each to make legs for some stools.

1. a) How many complete pieces of $35 \text{ cm}$ will he be able to get from the board?
2. b) How many millimeters of wood will he have left over after making the cuts?

The perimeter of a go-kart track measures $0,85 \text{ km}$. During a race, a driver has completed $12$ laps around the track.

Calculate the total distance traveled by the driver and express it in compound form (using kilometers, hectometers, and decameters).

A biologist is studying the growth of a root under a microscope. On the first day, the root measured $12 \text{ mm}$. After one week, the root measures $4,5 \text{ cm}$.

How many microns ($\mu m$) has the root grown in total during that week? (Remember that $1 \text{ mm} = 1.000 \mu m$).

To install fiber optic cable in a building, $3$ rolls of cable of $1,2 \text{ hm}$ each are needed. If the technician has already used $245 \text{ m}$ of cable, how many decameters of cable does he still have available?

An ant crawls along the edge of a rectangular table that measures $1,2 \text{ m}$ long and $80 \text{ cm}$ wide. If the ant completes $5$ full laps around the edge of the table:

1. a) How many meters does it travel in total?
2. b) Express the previous result in hectometers.

A hiking club is designing a new circular route in the Pyrenees. On the map (scale $1:25.000$), the route is divided into three sections: the first measures $12,4$ $cm$, the second $85$ $mm$, and the third $0,06$ $dm$. If a hiker walks at a constant speed and each of their steps measures exactly $75$ $cm$, calculate how many steps they will take to complete the entire route in reality. Also express the total distance of the route in hectometers ($hm$).

In a nanotechnology lab, a scientist observes a synthetic fiber whose thickness is $450$ nanometers ($nm$). For an experiment, he needs to place these fibers side by side until they cover a distance of $0,27$ millimeters ($mm$). Knowing that $1$ micrometer ($\mu m$) is $1.000$ $nm$ and that $1$ $mm$ is $1.000$ $\mu m$, how many fibers will the scientist have to place? If each fiber costs $\$0,002$, what would be the total cost of the sample?

In the past in Spain, the "vara castellana" was used, which is approximately equivalent to $83,59$ $cm$. A farmer reads in an old document that his land has a frontage of $120$ varas. To comply with current regulations, he must install a fence that costs $\$15,50$ per linear meter. However, for every $5$ decameters ($dam$) of fence, the store gives him a $10\%$ discount on the price of those meters. How much money will it cost him to fence the entire frontage of his land?

A racing drone flies around an irregular hexagon-shaped circuit. The lengths of its sides are: $0,15$ $km$, $2,4$ $hm$, $35$ $dam$, $1.200$ $dm$, $8.500$ $cm$, and $95.000$ $mm$. The drone must complete a total of $15$ laps of the circuit. What is the total distance the drone will travel in kilometers? If the drone's battery lasts long enough to travel $185$ hectometers, will it be able to finish all the laps or will it run out of battery beforehand? Justify your answer with calculations.

A math textbook has a total thickness (excluding the covers) of $2,4$ $cm$. If the book has exactly $400$ pages ($200$ sheets), what is the thickness of a single sheet expressed in microns ($\mu m$)? Knowing that a human hair has an average thickness of $0,08$ $mm$, how many human hairs would need to be stacked to equal the thickness of a single sheet of this book?

A tailor needs to edge $12$ identical circular tablecloths with a decorative silk ribbon. The circumference of each tablecloth is $37,5$ decimeters ($dm$). The ribbon is sold in rolls of $0,5$ hectometers ($hm$).

1. a) What is the minimum number of complete rolls the tailor must buy so that he does not run out of ribbon?
2. b) Once the $12$ tablecloths are finished, how many centimeters of ribbon will he have left over in total?
3. c) If each meter of ribbon costs $\$1,20$, what is the price of a complete roll?

The distance between two cities, A and B, is $145,6$ $km$. At the exact midpoint of the road connecting them, there is a sign indicating the distance to a third city C, located $34.500$ meters from that midpoint. If a car travels from A toward B and stops when it has traveled $820$ hectometers ($hm$):

1. a) How far is it from city B?
2. b) How far is it from the sign indicating the way to city C?

Express all results in meters.

An artificial satellite orbits Earth at a constant altitude of $350$ $km$. However, due to friction with the atmosphere, it loses $450$ meters of altitude each day.

1. a) At what altitude in kilometers will the satellite be after $3$ weeks in orbit?
2. b) If the satellite stops working when it drops below $330$ $km$ in altitude, how many full days will it be able to remain operational?
3. c) Express the initial altitude of the satellite in a compound expression using kilometers, hectometers, and meters.