The height of an open box is 1cm more than the length of a side of its square base.If the open box has a surface area of 96cms,find the dimensions using quadratic equation

If the base has side x, then

x^2 + 4x(x+1) = 96

To find the dimensions of the open box, we can first set up equations using the given information.

Let's assume the side length of the square base is "x" cm.

According to the problem, the height of the box is 1 cm more than the length of a side of its square base. So, the height will be x + 1 cm.

The surface area of the open box is given as 96 cm².

To break down the surface area, we need to consider the 6 faces of the box:

1. The top face has an area equal to the square base, which is x * x = x² cm².
2. The bottom face also has an area equal to the square base, which is x² cm².
3. The other 4 faces are identical rectangles with dimensions x * (x + 1) cm.

Therefore, the total surface area can be expressed as:

x² + x² + 4(x * (x + 1)) = 96

Simplifying the equation:

2x² + 4x² + 4x = 96
6x² + 4x - 96 = 0

This is a quadratic equation of the form ax² + bx + c = 0, where:
a = 6
b = 4
c = -96

To solve this quadratic equation, you can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Substituting the values into the formula:

x = (-(4) ± √((4)² - 4(6)(-96))) / (2(6))

x = (-4 ± √(16 + 2304)) / 12

x = (-4 ± √2320) / 12

Now, simplify the expression under the square root:

√2320 ≈ 48.17

Substitute this value back into the equation:

x = (-4 ± 48.17) / 12

x₁ = (-4 + 48.17) / 12 ≈ 3.18
x₂ = (-4 - 48.17) / 12 ≈ -5.85

Since the dimensions cannot be negative, we discard the negative value.

Therefore, the side length of the square base is approximately 3.18 cm.

To find the height, we can use the equation given earlier:

height = x + 1 = 3.18 + 1 = 4.18 cm

So, the dimensions of the open box are approximately 3.18 cm (side length of the square base) and 4.18 cm (height).