The height of a box is 10cm, its upper and lower lids are square in shape. If the total surface area of the box is 250 cm^2, the length of its side is :

Let's denote the side length of the square lids as "s" (in cm).

The box has two square lids, one at the top and one at the bottom, and four rectangular sides (since the height is 10 cm, and the other two sides are also squares). The total surface area (A) of the box is given as 250 cm².

We can calculate the total surface area of the box as follows:

1. The area of the top lid is s².
2. The area of the bottom lid is s².
3. The area of each of the four rectangular sides is s * 10 (height times side length).

So, the total surface area is:

A = 2 * (s²) + 4 * (s * 10)

A = 2s² + 40s

We know that the total surface area A is equal to 250 cm², so we can set up the equation:

2s² + 40s = 250

Now, let's solve this quadratic equation for "s." First, subtract 250 from both sides:

2s² + 40s - 250 = 0

Now, we can simplify the equation further by dividing all the terms by 2 to make it easier to solve:

s² + 20s - 125 = 0

Now, we can use the quadratic formula to find the value of "s":

s = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 1, b = 20, and c = -125. Plug these values into the formula:

s = (-20 ± √(20² - 4 * 1 * (-125))) / (2 * 1)

s = (-20 ± √(400 + 500)) / 2

s = (-20 ± √900) / 2

s = (-20 ± 30) / 2

Now, consider both the positive and negative roots separately:

1. Positive root:

s = (10) / 2 = 5 cm

1. Negative root (discard since length cannot be negative):

s = (-50) / 2 = -25 cm (discard)

So, the length of one side of the square lid is 5 cm.