5. (15 points) The density of a body $\rho$ is defined as the ratio of the body's mass $m$ to its volume $V$. A unit of mass used in jewelry is the carat (1 carat equals 0.2 grams). A unit of length used in many countries is the inch (1 inch equals 2.54 centimeters). It is known that the density of diamond is $\rho=3.5 \mathrm{r} / \mathrm{c}^{3}$. Convert this value to carats per cubic inch.
Answer: $\approx 287$ carats/inch ${ }^{3}$.

Solution. 1 gram $=\frac{1}{0.2}$ carat $=5$ carats,

1 cm $=\frac{1}{2.54}$ inch.

We get: $\rho=3.5 \frac{\mathrm{r}}{\mathrm{cm}^{3}}=3.5 \frac{5 \text { carats }}{\left(\frac{1}{2.54} \text { inch }\right)^{3}}=3.5 \cdot 5 \cdot 2.54^{3} \frac{\text { carats }}{\text { inch }{ }^{3}} \approx 287 \frac{\text { carats }}{\text { inch }{ }^{3}}$.

(5 points)