91Ó°ÊÓ

Cubic functions, as their name suggests, are polynomial functions of the third degree. They take the general form \( ax^3 + bx^2 + cx + d \), where \( a eq 0 \). These functions can model a variety of real-world situations involving growth, volume, and more.

One of the key features of cubic functions is that they can have up to three real roots or solutions. They can vary in the curvature and can have zero, one, or two turning points—places where the graph changes direction. In our specific optimization problem, the volume of the box is represented by the cubic function \( V(x) = 4x^3 - 280x^2 + 4000x \). This function helps us determine how changes in the value of \( x \) affect the volume of the box.

The key to solving our problem lies in finding the critical points of this cubic function, which helps us identify maximum or minimum volume points. By taking derivatives, we can determine where the slope of the function is zero, thus pinpointing these critical points.

Derivatives play a crucial role in calculus, particularly when it comes to optimization problems like ours. A derivative represents the rate of change of a function concerning one of its variables. In simpler terms, it shows how a function changes as its input changes. For a function \( f(x) \), the derivative is often represented as \( f'(x) \) or \( \frac{df}{dx} \).

In our exercise, we used the derivative to find the critical points of the cubic function representing the box's volume. We found that the derivative of \( V(x) = 4x^3 - 280x^2 + 4000x \) is \( V'(x) = 12x^2 - 560x + 4000 \). Setting this equal to zero, we look for values of \( x \) where the change in volume (or slope) is zero, which indicates potential maximum or minimum points.

Finding these critical points helps determine the optimal dimensions of the box, allowing us to choose a value of \( x \) that provides the maximum volume. By understanding derivatives, we are equipped with a powerful tool to solve problems involving change and maximize or minimize quantities efficiently.

To solve quadratic equations of the form \( ax^2 + bx + c = 0 \), the quadratic formula is a universal tool. It allows us to find the roots of the equation easily. The formula is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

The part under the square root, \( b^2 - 4ac \), is known as the discriminant. It tells us about the number and nature of the roots. If it’s positive, the equation has two distinct real roots; if zero, one real root; if negative, no real roots, only complex ones.

In our exercise, we solve the quadratic equation \( 3x^2 - 140x + 1000 = 0 \) derived from the derivative of the cubic volume function. Using the quadratic formula, we calculated:

• \( b^2 - 4ac = 7600 \), a positive discriminant indicating two real roots.
• Substituting the values, we find \( x = \frac{140 \pm 87.18}{6} \).

This computation gives us two possible solutions for \( x \): \( 37.9 \) and \( 8.8 \). Given the constraints of the problem, only \( x = 8.8 \text{ cm} \) provides feasible dimensions, as the shape must stay within the original cardboard's limits.