A box (with no top) is to be constructed from a piece of cardboard of sides A and B by cutting out squares of length h from the corners and folding up the sides as in the figure below: A Suppose that the box height is h = 4 in. and that it is constructed using 113 in. of cardboard (i.e., AB = 113). Which values A and B maximize the volume? A = in. B = in.
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.
![**Box Construction Problem**
A box (with no top) is to be constructed from a piece of cardboard with sides \( A \) and \( B \) by cutting out squares of length \( h \) from the corners and folding up the sides as shown in the figure below:
[Diagram description: The diagram shows a rectangle with dimensions \( A \) and \( B \). There are squares of side \( h \) cut from each corner. The remaining flaps are folded up to form the sides of the box.]
Suppose that the box height is \( h = 4 \) inches and it is constructed using 113 square inches of cardboard (i.e., \( AB = 113 \)). Which values of \( A \) and \( B \) maximize the volume?
\( A = \) [Input box] inches.
\( B = \) [Input box] inches.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9be8e2d-8356-4955-aeeb-a544f2c344f3%2F35c715f6-6845-4ea2-842a-fe45d1bdcfd8%2Fwvtvx55_processed.png&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 4 images
