Find the lengths of the sides of a rectangle that has an area of 144m2 and the smallest perimeter.

The task is given a rectangle with an area of ​​144 m². It is required to determine the lengths of the sides of a given rectangle so that the area of ​​the rectangle remains unchanged and its perimeter has the smallest value.
As you know, the area S of a rectangle with sides a and b is calculated by the formula S = a * b. Let’s substitute in this equality the given value of the area (without a unit of measurement): a * b = 144. Let’s express one of the sides, for example, b through the other (through a). We have: b = 144 / a.
Now let’s turn to the formula for calculating the perimeter P of a rectangle with sides a and b. It has the form: P = 2 * (a + b). Substitute here the expression b through a from the previous paragraph. Then, we get: P = 2 * (a + 144 / a). If we consider this equality as a function P (a) = 2 * a + 288 / a. That problem can be rephrased as follows: “Find such a value a> 0, which gives the smallest value of the function P (a) = 2 * a + 288 / a”. To solve this problem, we find the first derivative of the function P (a). We have РꞋ (а) = 2 – 288 / а². Equating the derivative to zero, we make the equation 2 – 288 / a² = 0. Let’s solve this equation. We have: 288 / a² = 2 or a² = 288/2 = 144, whence we get two different roots: a1 = -12 and a2 = 12.
By the condition of the problem, a> 0. Hence, the first root a = -12 must be discarded as a side one. Let, a = 12, then b = 144 / a = 144/12 = 12.
Thus, if the sides of a rectangle have equal values ​​(12 cm each), then such a rectangle (that is, a square) will have the smallest perimeter.