8. (15 points) Density is defined as the ratio of the mass of a body to its volume. There are two cubes. The second cube is made of a material with twice the density of the first, and the side length of the second cube is 100% greater than the side length of the first. By what percentage is the mass of the second cube greater than the mass of the first?
Answer: by $1500 \%$

Solution. From the condition, we get $\rho_{2}=2 \rho_{1}$, i.e., $\frac{m_{2}}{V_{2}}=2 \cdot \frac{m_{1}}{V_{1}}$.

Also, from the condition, it follows that the side length of the second cube $a_{2}=2 a_{1}$.

(2 points)

Therefore, their volumes are related by the ratio $V_{2}=a_{2}^{3}=\left(2 a_{1}\right)^{3}=8 V_{1}$.

(3 points)

We obtain $\frac{m_{2}}{8 V_{1}}=2 \cdot \frac{m_{1}}{V_{1}}$. (2 points) From which it follows that $m_{2}=16 m_{1}$.

That is, the mass of the second cube is greater by $1500 \%$.

![](https://cdn.mathpix.com/cropped/2024_05_06_31c3376e8d47665bbf90g-06.jpg?height=261&width=271&top_left_y=268&top_left_x=423)

Multidisciplinary Engineering Olympiad "Star" in Natural Sciences

6th grade

Final Stage

$2018-2019$

Variant 2

Tasks, answers, and evaluation criteria