Given a rectangle with sides of 3 cm and 5 cm If the larger side is reduced by a cm, and the smaller

Given a rectangle with sides of 3 cm and 5 cm If the larger side is reduced by a cm, and the smaller side is increased by a cm, then at what value, at what value is the forgiving of the resulting rectangle the greatest?

The area of a rectangle is found by the product of its length and width. By condition, the area of the rectangle can be expressed as a function of a:

S (a) = (5 – a) (3 + a) = 15 + 5a – 3a – a ^ 2 = – a ^ 2 + 2a + 15.

Let’s find the derivative of this function:

S ‘(a) = -2a + 2.

Let us find the extremum of this function, i.e. we equate the derivative function to zero:

-2a + 2 = 0;

Let’s solve this equation:

-2a = -2;

a = 1.

This point is the maximum of the function, therefore, for a = 1, the area of the rectangle will be maximum.

Answer: the area of the rectangle will be the largest at a = 1 cm.