Calculating the Dimensions of an Open-Top Box with a Square Base

In this guide, we will explore how to find the dimensions of an open-top box with a square base, given the height and the total surface area. This type of problem is common in geometry and can be useful for practical applications such as packaging design.

Understanding the Problem

We are given the following parameters:

Height of the box, (h 4) cm Total outer surface area of the box, (A 161) cm2

The goal is to determine the side length of the square base, (x).

Step-by-Step Solution

Let's break the problem into manageable steps.

Step 1: Surface Area Formula

The surface area, (A), of an open-top box with a square base can be calculated using the formula:

[A x^2 4(xh)]

In this formula:

(x^2) is the area of the base. (4(xh)) is the area of the four sides.

Step 2: Substitute Known Values

Substitute the given values into the formula:

[161 x^2 4(x cdot 4)]

This simplifies to:

[161 x^2 16x]

Step 3: Rearrange the Equation

Rearrange the equation to form a standard quadratic equation:

[x^2 16x - 161 0]

Step 4: Solve the Quadratic Equation

We solve this quadratic equation using the quadratic formula:

[x frac{-b pm sqrt{b^2 - 4ac}}{2a}]

Where (a 1), (b 16), and (c -161). First, calculate the discriminant:

[b^2 - 4ac 16^2 - 4 cdot 1 cdot (-161) 256 644 900]

Substitute the values into the quadratic formula:

[x frac{-16 pm sqrt{900}}{2 cdot 1} frac{-16 pm 30}{2}]

Calculate the two potential solutions:

[x frac{14}{2} 7 , text{cm}] [x frac{-46}{2} -23 , text{cm}]

In this context, only the positive root is valid since dimensions cannot be negative.

Conclusion

The length of one side of the square base is (7 , text{cm}).

Therefore, the dimensions of the base of the box are:

[7 , text{cm} times 7 , text{cm}]

To verify, let's factorize the quadratic equation:

[x^2 16x - 161 0 , text{can be factored as} , (x 23)(x - 7) 0]

This confirms the positive root, (x 7 , text{cm}).