rectangular box is to have a square base and a volume of 45 ft3. If the material for the base costs $0.19 per square foot, the material for the sides costs $0.09 per square foot, and the material for the top costs $0.11 per square foot, determine the dimensions of the box tha
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.
A rectangular box is to have a square base and a volume of 45 ft3. If the material for the base costs $0.19 per square foot, the material for the sides costs $0.09 per square foot, and the material for the top costs $0.11 per square foot, determine the dimensions of the box that can be constructed at minimum cost.
![### Understanding Rectangular Prisms: Calculating Dimensions
In this section, we'll learn how to determine the dimensions of a rectangular prism.
#### Diagram Explanation:
The image above presents a rectangular prism, commonly referred to as a box. The dimensions of this prism are labeled as follows:
- **Length (x)**: The horizontal distance from left to right.
- **Height (y)**: The vertical distance from the bottom to the top.
- **Width (x)**: The measurement identical to the length in the given orientation, thus simplifying our calculations.
#### Interactive Learning:
You are tasked with identifying the values of \( x \) and \( y \). Here are the fields provided:
- \( x = \) [_____ ] ft
- \( y = \) [_____ ] ft
Once you've determined the appropriate values for these dimensions, enter them into the corresponding input boxes.
#### Need Assistance?
If you require additional help or resources, make use of the following options:
- **Read It**: Access textual explanations and relevant theory.
- **Master It**: Engage with interactive exercises to deepen your understanding.
- **Talk to a Tutor**: Connect with a tutor for personalized assistance.
For more in-depth guidance or any queries, feel free to click on the help options at the bottom. Ensure to understand each part thoroughly for a robust grasp of calculating dimensions of geometric figures!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2860fba-712a-4be3-b6c0-f7c72a6e8972%2F338b8be9-c2fb-4355-92d7-8821ef47f67b%2Fy17dme7.jpeg&w=3840&q=75)
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