B4. A circular sector with an area of $375 \pi \mathrm{cm}^{2}$ and a radius of $25 \mathrm{~cm}$ is to be the lateral surface of a truncated cone. Calculate the surface area and volume of the solid accurately.

Space for solving problems in set $B$.

## dMFA

10th competition in mathematics knowledge for students of secondary technical and vocational schools

Regional competition, March 31, 2010

## PROBLEMS FOR FOURTH YEAR STUDENTS

Time for solving: 90 minutes. In set A, a correct answer will be worth two points, while a wrong answer will result in a deduction of half a point. Write the answers for set A in the left table.
![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-08.jpg?height=162&width=1194&top_left_y=632&top_left_x=448)
B4. We use the formula for the lateral surface area of a cone $S_{pl}=\pi r s$. We note that the side of the cone is equal to the radius of the circular sector $s=25 \mathrm{~cm}$. We calculate the radius of the cone $r=\frac{S_{pl}}{\pi}=15 \mathrm{~cm}$.

We use the formula for the surface area of the cone $P=\pi r^{2}+S_{pl}$ and calculate $P=600 \pi \mathrm{cm}^{2}$. We use the Pythagorean theorem to calculate the height of the cone $v^{2}=s^{2}-r^{2} . v=20 \mathrm{~cm}$. We use the formula for the volume $V=\frac{\pi r^{2} v}{3}=1500 \pi \mathrm{cm}^{3}$.

Conclusion that the radius of the sector is the side of the cone $s=25 \mathrm{~cm} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ point

Calculation of the radius of the cone $r=\frac{S_{pl}}{\pi}=15 \mathrm{~cm} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 1$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-18.jpg?height=66&width=1687&top_left_y=527&top_left_x=230)

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-18.jpg?height=65&width=1687&top_left_y=573&top_left_x=230)

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-18.jpg?height=54&width=1685&top_left_y=630&top_left_x=231)

Calculation of the volume $V=\frac{\pi r^{2} v}{3}=1500 \pi \mathrm{cm}^{3} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots$ point

NOTE: If the results are not accurate, one point is deducted.

## Fourth Year

## Group A

| A1 | A2 | A3 | A4 | A5 | A6 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| C | C | C | D | A | A |