Given a rectangle with sides of 8 cm and 12 cm. Its larger side was reduced by a cm

Given a rectangle with sides of 8 cm and 12 cm. Its larger side was reduced by a cm, and the smaller one was increased by a cm. At what value of a, the area of the rectangle will increase.

1. When changing the length and width of the rectangle, the area will also change, while it is not known whether it will become larger or smaller.
2. Determine the value of the changed length:
8 + a (cm);
3. New width value:
12 – a (cm);
4. Let’s find what the area of ​​the rectangle was equal to:
S = a * b;
8 * 12 = 96 (cm2);
5. Let’s find a new area S ‘:
(8 + a) * (12 – a) = 96 + 12a – 8a – a ^ 2 = -a ^ 2 + 4a + 96;
6. For the area to increase, the inequality must work:
S ‘/ S> 1;
(-a ^ 2 + 4a + 96) / 96> 1;
-a ^ 2 + 4a + 96> 96;
-a ^ 2 + 4a> 0;
a ^ 2 – 4a <0;
a * (a – 4) <0;
a ∈ (0; 4);
That is, the area will increase for any value of a from this interval. Of integers, these are: 1, 2, 3.
Answer: the area will increase for a ∈ (0; 4).