The perimeter of a rectangle is $18.6 \mathrm{~km}$, and its area is $15.62 \mathrm{~km}^{2}$. Find its length and width.

Step 1 ：We are given that the perimeter of a rectangle is 18.6 km and its area is 15.62 km². We can use these two pieces of information to find the length and width of the rectangle.

Step 2 ：The perimeter of a rectangle is given by the formula \(2*(length + width)\). We can set up an equation with this formula: \(2*(length + width) = 18.6\).

Step 3 ：The area of a rectangle is given by the formula \(length * width\). We can set up another equation with this formula: \(length * width = 15.62\).

Step 4 ：We now have a system of two equations. However, this system is a quadratic equation, so we will need to use the quadratic formula to solve it.

Step 5 ：Let's denote the coefficients in the quadratic formula as a, b, and c. In our case, a = 1, b = -9.3, and c = 15.62.

Step 6 ：We calculate the discriminant D in the quadratic formula, which is \(D = b^2 - 4ac\). Substituting the values of a, b, and c, we get \(D = 24.010000000000012\).

Step 7 ：Using the quadratic formula, we find two possible values for the length of the rectangle: \(length1 = 7.100000000000001\) km and \(length2 = 2.1999999999999997\) km.

Step 8 ：Similarly, we find two possible values for the width of the rectangle: \(width1 = 2.1999999999999993\) km and \(width2 = 7.100000000000001\) km.

Step 9 ：However, since the length of a rectangle is usually considered to be the longer side, we take the larger value as the length and the smaller value as the width.

Step 10 ：\(\boxed{\text{Final Answer: The length and width of the rectangle are approximately 7.1 km and 2.2 km respectively.}}\)