Question

Density

which weighs more, a ball of zinc with a radius of 3 cm, or a ball of chromium with a radius of 2.99 cm

Answer 1

To determine which ball weighs more, we need to know the density of zinc and chromium. The density of zinc is **7.14 g/cm³** and the density of chromium is **7.19 g/cm³**. The volume of a sphere is given by the formula `V = (4/3)πr³`, where `r` is the radius of the sphere. So, the volume of the zinc ball with a radius of 3 cm is `V = (4/3)π(3)³ ≈ 113.1 cm³`, and the volume of the chromium ball with a radius of 2.99 cm is `V = (4/3)π(2.99)³ ≈ 112.1 cm³`. The mass of an object is given by the formula `m = ρV`, where `ρ` is the density of the material and `V` is the volume of the object. So, the mass of the zinc ball is `m = (7.14 g/cm³)(113.1 cm³) ≈ 807.6 g`, and the mass of the chromium ball is `m = (7.19 g/cm³)(112.1 cm³) ≈ 806.1 g`. Therefore, **the ball of zinc weighs more** than the ball of chromium.

Answer 2

A ball of zinc.

Explanation:

Density is defined as the ratio of an object's mass to its volume.

The formula for density is:

`\large\boxed{\rho = (m)/(V)}`

where:

• ρ is density measured in kilograms per cubic metre (g/cm³).
• m is mass measured in grams (g).
• V is volume measured in cubic centimeters (cm³).

Therefore, to calculate the weight of each ball, multiply its volume by the density of the material.

A ball can be modelled as a sphere. Therefore, calculate the volumes of the balls by using the formula for the volume of a sphere.

`\boxed{\begin{minipage}{4 cm}\underline{Volume of a sphere}\\\\$V=(4)/(3) \pi r^3$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}`

Substitute the respective values of r into the formula to calculate the volumes of the two balls.

`\begin{aligned}\textsf{Volume of the ball of zinc}&=(4)/(3) \pi (3)^3\\\\&=(4)/(3) \pi (27)\\\\&=36\pi\\\\&=113.0973355292... \; \sf cm^3\end{aligned}`

`\begin{aligned}\textsf{Volume of the ball of chronium}&=(4)/(3) \pi (2.99)^3\\\\&=(4)/(3) \pi (26.730899)\\\\&=35.6411986666...\pi\\\\&= 111.9701278963...\; \sf cm^3\end{aligned}`

The density of zinc (Zn) is approximately 7.14 g/cm³.

The density of chromium (Cr) is approximately 7.15 g/cm³.

Therefore, the approximate weights of the two balls are:

`\begin{aligned}\textsf{Weight of the ball of zinc}&=113.0973355292... * 7.14\\&=807.514975...\\&=807.51 \; \sf g\end{aligned}`

`\begin{aligned}\textsf{Weight of the ball of chronium}&= 111.9701278963... * 7.15\\&=800.586414...\\&=800.59\; \sf g\end{aligned}`

As 807.51 > 800.59, a ball of zinc with a radius of 3 cm weighs more than a ball of chromium with a radius of 2.99 cm.