Real objects are not perfect cylinders or boxes, but they are often close. Modeling means choosing a shape that fits well enough, using the volume and area tools from Surface Area of Solids and Scaling Area and Volume, and then checking whether the answer makes sense.
- Choose a model: a tree trunk as a cylinder, a scoop of ice cream as a sphere, a roof as a prism.
- Measure or estimate the dimensions, with units.
- Compute, keeping the units visible in every step.
- Check: is the size reasonable? Would a different model change the answer much?
Why it works
Density is an amount per unit of space: grams per cubic centimeter, people per square mile, plants per square foot. Because it is a ratio, you can rearrange it: $\text{amount} = \text{density} \times \text{size}$. Writing the units as fractions shows what cancels, which is the safest way to convert between units.
Key idea
Model the object with a shape, compute with units attached, and use density as "amount per unit of length, area or volume." Convert units by multiplying by fractions equal to 1, like $\frac{1 \text{ mi}^2}{2.59 \text{ km}^2}$.
Worked example: Mass from density
An aluminum block is $5$ cm by $4$ cm by $3$ cm. Aluminum's density is $2.7$ g/cm$^3$. Find its mass.
- Volume: $5 \cdot 4 \cdot 3 = 60$ cm$^3$.
- Mass $= 2.7 \frac{\text{g}}{\text{cm}^3} \cdot 60 \text{ cm}^3 = 162$ g.
Answer: $162$ g
Worked example: Population density and units
A city has $1{,}200$ people per square mile. About how many people per square kilometer is that? ($1$ mi$^2 \approx 2.59$ km$^2$.)
- ${}1{,}200 \frac{\text{people}}{\text{mi}^2} \cdot \frac{1 \text{ mi}^2}{2.59 \text{ km}^2} \approx 463 \frac{\text{people}}{\text{km}^2}$.
- Check: a square kilometer is smaller than a square mile, so fewer people fit in each one.
Answer: About $463$ people per km$^2$
Worked example: A design question
A company ships $36$ cylindrical cans, each with diameter $3$ in and height $5$ in, upright in a box that is $6$ cans by $6$ cans in one layer. How much of the box's volume do the cans fill?
- Box: $18 \times 18 \times 5 = 1620$ in$^3$.
- Cans: $36 \cdot \pi \cdot 1.5^2 \cdot 5 = 405\pi \approx 1272.3$ in$^3$.
- Fraction: $\frac{405\pi}{1620} = \frac{\pi}{4} \approx 0.785$, about $79\%$.
Answer: About $79\%$ (the same for any square packing of cylinders: $\frac{\pi}{4}$)
Common mistake
Mixing units, like dividing people by square kilometers when the area is in square miles, or converting areas with the length factor. $1$ mi $\approx 1.61$ km, but $1$ mi$^2 \approx 1.61^2 \approx 2.59$ km$^2$. Write units in every step; if they do not cancel to the units you want, something is off.