Length and Its Units

In the study of mathematics and physics, it is essential to distinguish between what we measure and the unit we use. Classify the following concepts in your notebook as either 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 suitable unit. State which metric unit ($mm$, $cm$, $m$ 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 a latest-generation smartphone screen.

The metric system is a base-10 decimal system because each unit is $10$ times larger 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 measuring 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) $0.5\text{ km} = \dots\dots\dots\text{ hm}$

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

1. a) How many full pieces of $35 \text{ cm}$ can he get from the batten?
2. b) How many millimeters of wood will he have left over after making the cuts?

Three friends compare the length of their headphone cables to see which one is longer. The measurements are:

• Marcos: $1.2\text{ m}$

• Lucía: $125\text{ cm}$

• Hugo: $11.5\text{ dm}$

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

A stage of a hiking route through the Sierra Madre 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 is the complete stage?

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

Calculate the total distance of the journey, expressing the result as a single unit in kilometers.

The distance between two cities, A and B, is $145.6$ $km$. Exactly at the midpoint of the road connecting them, there is a signpost indicating the distance to a third city C, located $34,500$ meters from that midpoint. If a car travels from A towards B and stops after covering $820$ hectometers ($hm$):

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

Express all results in meters.

Sara 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 add a protective border of $5\text{ mm}$ to 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 five-a-side soccer field measuring $40\text{ m}$ in length and $200\text{ dm}$ in width. Express the final result in meters.

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

The perimeter of a go-kart track measures $0.85 \text{ km}$. During a race, a driver completed $12$ laps of the track.

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

An ant walks along the edge of a rectangular table measuring $1.2 \text{ m}$ 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 cover in total?
2. b) Express the previous result in hectometers.

In a tech shop, monitor screens are measured by the length of their diagonal in inches. We know that $1 \text{ pulgada} = 2.54 \text{ cm}$.

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

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 micrometres ($\mu m$) has the root grown in total during that week? (Remember that $1 \text{ mm} = 1,000 \mu m$).

In a nanotechnology laboratory, a scientist observes a synthetic fiber whose thickness is $450$ nanometers ($nm$). For an experiment, they need to place these fibers side by side to 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 need to place? If each fiber costs $0.04$ pesos, what would be the total cost of the sample?

Three friends are comparing the distances they ran this morning while training for a charity race. Lucía says she ran $4.2 \text{ km}$, Marcos claims to have covered $38 \text{ hm}$ and $15 \text{ dam}$, and Sofía recorded a total of $4,050 \text{ m}$ on her smartwatch.

Order the three friends from greatest to shortest distance covered, expressing all measurements in meters to compare them.

A racing drone completes a circuit shaped like an irregular hexagon. 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 cover $185$ hectometers, will it be able to finish all the laps or will it run out of battery first? Justify your answer with calculations.

An artificial satellite orbits the Earth at a constant altitude of $350$ $km$. However, due to atmospheric drag, 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 falls below $330$ $km$ in altitude, for how many full days will it remain operational?
3. c) Express the satellite's initial altitude as a compound expression using kilometers, hectometers, and meters.