2. Agronomist Bilbo noticed that if the length of his rectangular field were 20 meters longer, the perimeter of the field would be twice as large. However, if the width of the field were twice as large, the perimeter of the field would be 18 meters larger. What is the area of the field?

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# Answer: $99 \mu^{2}$.

## Solution:

## Method № 1.

If the width of the field has doubled, then the perimeter has simply increased by twice the width of the field, but according to the problem, this change is 18 meters - which means the width of the field is 9 meters. When the length of the field increases by 20 meters, the perimeter of the field increases by 40 meters, but (according to the problem) this change is equal to the perimeter of the field (since the perimeter has doubled). Therefore, the perimeter of the field is 40 meters. Then the length and width together make 20 meters, and the width is 9 meters, so the length is 11 meters, and the area is $9 \cdot 11 = 99\left(\mu^{2}\right)$.

## Method № 2.

Essentially the same, but with symbolic notation. Let the length of the field be $x$, and the width of the field be $y$. Then the perimeter is $2x + 2y$. When the length of the field increases by 20 meters, we get a rectangle with length $x + 20$ and width $y$, its perimeter is $2(x + 20) + 2y$, which, according to the problem, equals $2(2x + 2y)$.

We get the equation: $2(x + 20) + 2y = 2(2x + 2y)$, from which $2x + 2y = 40$, then $x + y = 20(m)$.

When the width of the original field is doubled, we get a rectangular field with length $x$, width $2y$, and perimeter $2x + 4y$, which, according to the problem, equals $2x + 2y + 18$.

From the equation $2x + 4y = 2x + 2y + 18$ we get that $2y = 18, y = 9$ (meters).

Then $x = (x + y) - y = 20 - 9 = 11$ (meters), and the area is $9 \cdot 11 = 99\left(\mu^{2}\right)$.

## Criteria:

A correct answer without justification and without an example of suitable length and width - give 1 point. A correct answer without justification, but with an indication of the length and width at which it is obtained, give 2 points.