B4 Posoda, in which we grow seedlings, is made of wood and covered with glass (see picture). The thickness of the boards and glass is not considered in the calculations.

A What is the area of the top glass surface?

B By how many $d m^{2}$ is the area of the wooden bottom less than the area of the glass cover?

![](https://cdn.mathpix.com/cropped/2024_06_07_8c4ca9fb11cdb9201aa3g-3.jpg?height=423&width=671&top_left_y=2144&top_left_x=1246)

C How many liters of soil do we need to fill the pot to half its height?

## 11th Mathematics Knowledge Competition for Vocational School Students

Regional Competition, March 30, 2011

## Solutions to Problems and Scoring

A contestant who arrives at the solution by any correct method (even if the scoring does not provide for it) receives all possible points.

For a correct method, we consider any procedure that

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

$V$ in the table, the correct answers to the multiple-choice questions are recorded. Each correct answer is scored 2 points, an incorrect answer -0.5 points, if the question is not answered, 0 points. To avoid a possible negative final score, each contestant is given an initial 5 points.

| A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | A10 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| C | A | D | B | E | B | C | C | D | D |
B4 The area of the upper rectangle (glass surface) is $60 \mathrm{~cm} \cdot 36 \mathrm{~cm}=2160 \mathrm{~cm}^{2}$. The area of the lower rectangle (wooden bottom) is $60 \mathrm{~cm} \cdot 20 \mathrm{~cm}=1200 \mathrm{~cm}^{2}$. The difference between the two areas is $2160 \mathrm{~cm}^{2}-1200 \mathrm{~cm}^{2}=960 \mathrm{~cm}^{2}=9.6 \mathrm{dm}^{2}$. To calculate how many liters of soil are needed to fill the pot to half its height, we need to calculate the volume of the prism with height $v_{p}=60 \mathrm{~cm}$, which has a trapezoid as its base with bases $a=20 \mathrm{~cm}$ and $c=28 \mathrm{~cm}$ and height $v_{t}=8.5 \mathrm{~cm}: V=\frac{(a+c) \cdot v_{t}}{2} \cdot v_{p}=\frac{(20+28) \cdot 8.5}{2} \cdot 60=12240 \mathrm{~cm}^{3}=$ $12.24 \mathrm{dm}^{3}=12.24 \mathrm{L}$.

A Answer, e.g.: The area of the upper glass surface is $2160 \mathrm{~cm}^{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ t

B Answer, e.g.: The area of the wooden bottom is smaller than the area of the glass cover by $9.6 \mathrm{dm}^{2}$. $2 \mathrm{t}$

Note: If the result is not expressed in $dm^{2}$, we deduct one point from the contestant.

![](https://cdn.mathpix.com/cropped/2024_06_07_8c4ca9fb11cdb9201aa3g-7.jpg?height=65&width=1642&top_left_y=1555&top_left_x=264)

Justification of the answer with calculation. ......................................................... 1 t