## Task B-3.7.

A right triangle $ABC$, with the leg $|BC|=10$ and the opposite angle $\alpha=30^{\circ}$, rotates around the line $p$ passing through vertex $A$ such that it forms a $30^{\circ}$ angle with the line $AB$ ($C \notin p$). What is the volume of the rotational body thus formed?
## Solution.

By rotation, a body is formed consisting of a cone $S_{1}$ and a frustum of a cone with a common (larger) base, and without the cone $S_{2}$ which has a common smaller base with the frustum.

Sketch or description

The volume of the resulting body will be calculated by adding the volume of the cone $S_{1}$ and the volume of the frustum, and then subtracting the volume of the cone $S_{2}$.
![](https://cdn.mathpix.com/cropped/2024_05_30_6dc7744662d888032875g-22.jpg?height=734&width=1466&top_left_y=1261&top_left_x=223)

Notice the congruent triangles: $\triangle A B C \cong \triangle A B D$ (ASA - they have congruent angles and a common side $\overline{A B}$.

The lengths of the sides of triangle $A B C$ are:

$$
a=|B C|=10, \quad b=|A C|=a \operatorname{ctg} 30^{\circ}=10 \sqrt{3}, \quad c=|A B|=\frac{a}{\sin 30^{\circ}}=20
$$

The radius of the smaller base of the frustum is $r=a=10$.

From triangle $A C E$, we calculate the radius of the larger base of the frustum:
$R=b \sin 60^{\circ}=10 \sqrt{3} \cdot \frac{\sqrt{3}}{2}=15$.

1 point

In the same triangle, the height of the cone $S_{1}$ is $v_{1}=|A E|=b \cos 60^{\circ}=5 \sqrt{3}$.

1 point

The height of the cone $S_{2}$ is $v_{2}=|A D|=b=10 \sqrt{3}$.

1 point

The height of the frustum is $v=v_{2}-v_{1}=5 \sqrt{3}$.

1 point

Let's calculate the desired volume.

$$
\begin{aligned}
V & =\frac{1}{3} R^{2} \pi v_{1}+\frac{1}{3} \pi v\left(R^{2}+R r+r^{2}\right)-\frac{1}{3} r^{2} \pi v_{2} \\
& =\frac{1}{3} \pi(1125 \sqrt{3}+2375 \sqrt{3}-1000 \sqrt{3})=\frac{2500 \pi \sqrt{3}}{3}
\end{aligned}
$$

$$
1 \text { point }
$$

1 point

## COUNTY COMPETITION IN MATHEMATICS

## 4th grade - high school - B variant

## February 28, 2018.

## IF A STUDENT HAS A DIFFERENT METHOD OF SOLVING THE PROBLEM, THE COMMITTEE IS OBLIGED TO GRADE AND EVALUATE THAT METHOD APPROPRIATELY.