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Choosing the solid that stands for a real object, and computing density, surface area and population density from it.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you learn to model a real object with the solid that is close enough to it: a tin as a cylinder, a tank or a parcel as a box, a town as a rectangle. Once the shape is chosen, the volume and area formulas you already have do the work. You use them to find density, which is mass divided by volume, population density, which is people divided by area, and the surface area a label, a coat of paint or a sheet of metal has to cover — and you learn to tell which of those a question is really asking for.
You can find the volume of a box and of a cylinder, the area of a rectangle and of a circle, and you can divide one number by another. Nothing here is a new formula. What is new is that nobody hands you the shape: you look at a real object and decide which solid is close enough to stand for it.
Model: a shape chosen to stand for a real object, on purpose imperfect. A tin is not exactly a cylinder, but treating it as one gets an answer that is right to within the accuracy anybody needs.
Density: mass per unit of volume, written in grams per cubic centimetre or kilograms per cubic metre.
Population density: people per unit of area, usually per square kilometre.
Per: the word that means divide. Grams per cubic centimetre is grams $\div$ cubic centimetres; people per square kilometre is people $\div$ square kilometres.
Curved surface area of a cylinder: the tube without its two ends — the part a label covers.
Surface area: the total area of every face, which is what a covering, a coat of paint or a sheet of metal has to reach.
Real objects are not solids from a textbook, so you choose the solid that is close enough and work with that. A tin of soup is a cylinder; a brick, a shipping container or a room is a box; a marble or a water droplet is a sphere; a pile of sand is a cone. Once the shape is chosen the formulas you already know take over, and the answer is only as good as the model — which is usually good enough. Two of the most useful quantities are rates, and both come from dividing: density is mass $\div$ volume, so a block $10$ by $5$ by $4$ cm has volume $200$ cm$^3$ and a mass of $1600$ g gives $8$ g per cm$^3$. Population density is people $\div$ area, so $120\,000$ people on a town $8$ km by $5$ km is $120\,000 \div 40 = 3000$ people per km$^2$. Surface area answers the covering questions: the curved side of a cylinder unrolls into a rectangle $2\pi r$ by $h$, so a tank of radius $2$ m and height $5$ m has $2 \times 3.14 \times 2 \times 5 = 62.8$ m$^2$ of curved surface, and a box $a$ by $b$ by $c$ has $2(ab + bc + ca)$.
Another way: picture
A tin of soup with a dashed cylinder drawn over it, radius and height labelled. Beside it the label lies flat: a plain rectangle, $2\pi r$ along the bottom and $h$ up the side.
Another way: story
Per means divide, every time. Grams per cubic centimetre is mass shared out over the volume; people per square kilometre is a population shared out over the land. Say the unit aloud and it tells you which number goes on top.
Dividing the wrong way round. Density is mass $\div$ volume. If a block of $200$ cm$^3$ has mass $1600$ g, the density is $8$ g per cm$^3$, not $0.125$. Read the unit out loud: grams per cubic centimetre names the top of the fraction first.
Reaching for volume when the question is about a covering. Paint, labels, wrapping and sheet metal are all area. Filling, holding and weighing are volume. Decide which one the object is doing before you pick a formula.
Including the ends of a cylinder when the question does not. A label round a tin covers $2\pi r h$ and nothing else. A closed can that has to be made out of metal also needs the two circles, $2\pi r^2$. The words label, closed and open are doing real work in the question.
Model the block as a box: volume $= 10 \times 5 \times 4 = 200$ cm$^3$.
Three lengths multiplied, not two.
Density $=$ mass $\div$ volume $= 1600 \div 200$.
Per means divide.
$= 8$ grams per cm$^3$.
Which is about right for a block of steel.
The label covers the curved side only, and unrolled it is a rectangle.
Its height is the tin's height; its width is the way round.
Round the tin is $2\pi r = 2 \times 3.14 \times 2 = 12.56$ cm.
Label $= 12.56 \times 9 = 113.04$ cm$^2$.
An area, so the unit is squared.
Population density is people $\div$ area.
$500\,000 \div 250 = 2000$ people per km$^2$.
A solid metal paperweight is close enough to a rectangular box: it measures $9$ cm by $6$ cm by $5$ cm, and its mass is $1620$ g. What is its density, in grams per cubic centimetre?
answer
The paper label on a tin of coffee wraps once round the curved side with no overlap, and covers neither end. Model the tin as a cylinder of radius $10$ cm and height $20$ cm, and take $\pi$ as $3.14$. How much paper does the label use? Give the number and its unit.
Answer: unit: m2 / cm2
A market town covers a patch of land close enough to a rectangle, $7$ km by $6$ km, and $130200$ people live in it. What is its population density, in people per square kilometre?
answer
A closed water tank is built as a rectangular box $5$ m by $6$ m by $2$ m. How many square metres of steel sheet does its outside take?
Answer:
A courier charges for a parcel by whichever is greater: its actual mass in kilograms, or its *volumetric weight*, which the company defines as the volume of the box in cubic centimetres divided by $5000$. A box measures $50$ cm by $60$ cm by $60$ cm and the thing inside it has a mass of $24$ kg. Which of the two will the courier charge for?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A solid metal paperweight is close enough to a rectangular box: it measures $5$ cm by $8$ cm by $3$ cm, and its mass is $360$ g. What is its density, in grams per cubic centimetre?
answer
You can choose a solid to stand for a real object and compute density, surface area and population density from it. Explain why a label round a tin does not need the area of its two ends, and say what the word per tells you to do.
8. Your turn: a city of $500\,000$ people covering $250$ km$^2$, step 2