A rectangular box with a volume of 980 ft is to be constructed with a square base and top. The cost per square foot for the bottom is 20g, for the top is 15g, and for the sides is 2.5g. What dimensions will minimize the cost? What are the dimensions of the box? The length of one side of the base is The height of the box is (Round to one decimal place as needed.)
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 with a volume of 980 ft³ is to be constructed with a square base and top. The cost per square foot for the bottom is 20¢, for the top is 15¢, and for the sides is 2.5¢. What dimensions will minimize the cost?
**What are the dimensions of the box?**
- The length of one side of the base is [ ]
- The height of the box is [ ]
(Round to one decimal place as needed.)
**Enter your answer in each of the answer boxes.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e346613-be2f-4a74-a0b2-1091c1f3df4a%2F7e88dfbe-db5e-418f-9bee-5647e1ea765e%2Fml2t8du_processed.jpeg&w=3840&q=75)
![A university is trying to determine what price to charge for tickets to football games. At a price of $20 per ticket, attendance averages 40,000 people per game. Every decrease of $2 adds 10,000 people to the average number. Every person at the game spends an average of $4.00 on concessions. What price per ticket should be charged in order to maximize revenue? How many people will attend at that price?
**What is the price per ticket?**
- $ [ ]
**What is the average attendance?**
- [ ] people
*Enter your answer in each of the answer boxes.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e346613-be2f-4a74-a0b2-1091c1f3df4a%2F7e88dfbe-db5e-418f-9bee-5647e1ea765e%2Ff27c929_processed.jpeg&w=3840&q=75)
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