A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 20 in. by 30 in. by cutting out equal squares of side x at each corner and then folding up the sides. Express the Surface area and volume of the box as a function of x.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
The dimensions of the cardboard is 20 in by 30 in.
To construct a box with open top, squares of side x at each corner has to be cut.
Since we are cutting x out of each corner, the length of the box must be (30-2x), width must be (20-2x) and the height is x.
Evaluate the volume of the box as follows.
Therefore, the volume of the box is .
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