A small company makes custom surfboards. They have found that if they sell the surfboards for $100 thedemand is 100 boards. If they sell the surfboards for $80, the demand is 140 surfboards.a. Find a linear demand equation showing the number of surfboards, q, demanded as a function of theselling price p and then use the demand function to build the revenue function.b. If the marginal cost to produce each surfboard is $40 and there are fixed costs of $1000, report thecost and profit functions as a function of the price p.c. Determine what value ofpwill maximize profit? What is the maximum profit? How many surfboardsare sold at that price?
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 small company makes custom surfboards. They have found that if they sell the surfboards for $100 the
demand is 100 boards. If they sell the surfboards for $80, the demand is 140 surfboards.
a. Find a linear demand equation showing the number of surfboards, q, demanded as a function of the
selling price p and then use the demand function to build the revenue function.
b. If the marginal cost to produce each surfboard is $40 and there are fixed costs of $1000, report the
cost and profit functions as a function of the price p.
c. Determine what value of
p
will maximize profit? What is the maximum profit? How many surfboards
are sold at that price?
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