5. Janja decided to raise chickens and ducks ecologically and intends to fence a rectangular plot with an area of $1632 \mathrm{~m}^{2}$. She will keep the ducks and chickens separate, so she plans to divide the plot in half along its width (see the image). For the fencing, she will need a total of 198 meters of fence. What is the width and length of the plot?

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-04.jpg?height=228&width=377&top_left_y=1665&top_left_x=1539)

(8 points)

## d MFA

12th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools

National Competition, April 21, 2012

## PROBLEMS FOR 4th GRADE

Time for solving: 120 minutes.

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-05.jpg?height=172&width=411&top_left_y=371&top_left_x=1502)
5. Let the length of the plot be denoted by $x$ and the width by $y$. The area of the plot (the area of the rectangle) is $x \cdot y=1632$. The fence consists of all 4 sides and the cross fence: $2 x+3 y=198$. We have obtained a system of equations, which is easiest to solve using the substitution method. From the first equation, we express, for example, $y=\frac{1632}{x}$ and substitute it into the second equation. We multiply by $x$, obtaining the quadratic equation $x^{2}-99 x+2448=0$. We get two solutions: $x_{1}=51\left(y_{1}=32\right)$ and $x_{2}=48\left(y_{2}=34\right)$.

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-12.jpg?height=55&width=1636&top_left_y=1189&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-12.jpg?height=58&width=1628&top_left_y=1236&top_left_x=208)

Solving the system of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $2^{*}$ points

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-12.jpg?height=52&width=1636&top_left_y=1339&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-12.jpg?height=45&width=1636&top_left_y=1391&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-12.jpg?height=48&width=1636&top_left_y=1438&top_left_x=210)

Writing the answer.............................................................................................

## Fourth Year