# Problem №4 (10 points)

A parallel beam of light falls on the base of a glass cone (refractive index $n=1.5$) along its axis (see fig.). The cross-section of the beam coincides with the base of the cone, the radius

![](https://cdn.mathpix.com/cropped/2024_05_06_167c0776c22cdf752babg-06.jpg?height=371&width=845&top_left_y=2407&top_left_x=1005)
of which is $R=1 \mathrm{~cm}$. The height of the cone is $h=1.73 \mathrm{~cm}$. Determine the area of the bright spot on the screen, perpendicular to the axis of the cone and located at a distance $a=1$ cm from the vertex of the cone.

#
# Solution:

![](https://cdn.mathpix.com/cropped/2024_05_06_167c0776c22cdf752babg-07.jpg?height=388&width=757&top_left_y=483&top_left_x=84)

$\operatorname{tg} \alpha=\frac{h}{R}=1.73$, i.e., $\alpha=60^{\circ}$

The law of refraction for rays passing through the cone:

$\frac{\sin \alpha}{\sin \beta}=\frac{1}{1.5}$

As a result: $\sin \beta=\frac{3 \sqrt{3}}{4} \geq 1$

We obtain that the rays passing through the cone experience total internal reflection.

![](https://cdn.mathpix.com/cropped/2024_05_06_167c0776c22cdf752babg-07.jpg?height=968&width=900&top_left_y=1709&top_left_x=87)

From geometric considerations, it is found that they fall on the other side of the cone at the normal.

(2 points)

Therefore, the light spot on the screen is bounded by two circles with radii:

$R_{1}=a \cdot \operatorname{tg} 60^{\circ}=0.01 \sqrt{3}=0.0173 \text{ m}$

$R_{2}=\left((h+a) \cdot \operatorname{tg} 60^{\circ}\right)-R=0.0273 \sqrt{3}-0.01=0.0373 \text{ m}$

We obtain:

$S=\pi R_{2}^{2}-\pi R_{1}^{2}=0.0034 \text{ m}^{2}=34 \text{ cm}^{2}$

# Multidisciplinary Engineering Olympiad "Star"  NATURAL SCIENCES  2016 

March 29, 2016

PROTOCOL № 1 of the meeting of the jury

PRESENT: Keller A.V., Zalyapin V.I., Zamishlyayeva A.A., Vorontsov A.G., Kuts D.A., Gusev A.V.

HEARD: on the distribution of points among the winners and prizewinners of the olympiad

## 11th Grade

- to consider the winners of the olympiad and award diplomas of the 1st degree to participants who scored 100 - 90 points;
- to consider the prizewinners of the olympiad and award diplomas of the 2nd degree to participants who scored 89 - 70 points;
- to consider the prizewinners of the olympiad and award diplomas of the 3rd degree to participants who scored 69 - 45 points.

## 10th Grade

- to consider the winners of the olympiad and award diplomas of the 1st degree to participants who scored 100 - 95 points;
- to consider the prizewinners of the olympiad and award diplomas of the 2nd degree to participants who scored 94 - 70 points;
- to consider the prizewinners of the olympiad and award diplomas of the 3rd degree to participants who scored 69 - 40 points.

## 9th Grade

- to consider the winners of the olympiad and award diplomas of the 1st degree to participants who scored 100 - 90 points;
- to consider the prizewinners of the olympiad and award diplomas of the 2nd degree to participants who scored 89 - 70 points;
- to consider the prizewinners of the olympiad and award diplomas of the 3rd degree to participants who scored 69 - 40 points.

## 8th Grade

- to consider the winners of the olympiad and award diplomas of the 1st degree to participants who scored 100 - 75 points;
- to consider the prizewinners of the olympiad and award diplomas of the 2nd degree to participants who scored 74 - 65 points;
- to consider the prizewinners of the olympiad and award diplomas of the 3rd degree to participants who scored 64 - 40 points.

## 7th Grade

- to consider the winners of the olympiad and award diplomas of the 1st degree to participants who scored 100 - 90 points;
- to consider the prizewinners of the olympiad and award diplomas of the 2nd degree to participants who scored 89 - 70 points;
- to consider the prizewinners of the olympiad and award diplomas of the 3rd degree to participants who scored 69 - 40 points.

## 6th Grade

- to consider the winners of the olympiad and award diplomas of the 1st degree to participants who scored 100 - 90 points;
- to consider the prizewinners of the olympiad and award diplomas of the 2nd degree to participants who scored 89 - 80 points;
- to consider the prizewinners of the olympiad and award diplomas of the 3rd degree to participants who scored 79 - 40 points.

## Chairmen of the jury:

![](https://cdn.mathpix.com/cropped/2024_05_06_167c0776c22cdf752babg-10.jpg?height=257&width=582&top_left_y=1096&top_left_x=180)

Keller A.V.

Vorontsov A.G.