The number of hours needed to paint a house is given by H(x)=320/x+x^2−6 if x people are working on painting the house. Here, a fraction of a person (i.e., if x is not an integer) represents a person working part-time. Use calculus to find the minimum length of time it takes to paint the house. Make sure to use calculus to verify that the time you find is a local minimum. The total cost to paint the house is the product of length of time, the number of people working, and the hourly salary ($50 / hour). For what number of workers is the total cost minimized?
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 number of hours needed to paint a house is given by H(x)=320/x+x^2−6 if x people are working on painting the house. Here, a fraction of a person (i.e., if x is not an integer) represents a person working part-time.
- Use calculus to find the minimum length of time it takes to paint the house. Make sure to use calculus to verify that the time you find is a
local minimum . - The total cost to paint the house is the product of length of time, the number of people working, and the hourly salary ($50 / hour). For what number of workers is the total cost minimized?
Step by step
Solved in 2 steps with 2 images
