The area of the rectangle is 36 dm2 what lengths should its sides have so that the perimeter of the rectangle

The area of the rectangle is 36 dm2 what lengths should its sides have so that the perimeter of the rectangle is the smallest a) set the function formula for which it will be necessary to find the minimum point b) find the lengths of the sides of the rectangle satisfying the condition of the problem

1. Let a and b be the sides of the rectangle. Then:

ab = S = 36, from here we get:
b = 36 / a.
2. Let’s make an equation for the perimeter of the rectangle:

P = 2 (a + b);
P = 2 (a + 36 / a).
3. Find the smallest value of the perimeter by calculating the derivative of the function P (a):

P (a) = 2 (a + 36 / a);
P ‘(a) = 2 (1 – 36 / a ^ 2);
2 (1 – 36 / a ^ 2) = 0;
1 – 36 / a ^ 2 = 0;
36 / a ^ 2 = 1;
a ^ 2 = 36;
1) a = -6, does not satisfy the condition of the problem;

2) a = 6 (dm);

b = 36 / a = 36/6 = 6 (dm).

Thus, we get the smallest perimeter value if the rectangle is a square.

Answer: 6 dm and 6 dm.