The amount of revenue a company makes per day by selling x items is given by the function f(x) = 14x - 0.2x2. How many items should be sold if the company wants to maximize their profit?
I suppose you mean f(x) = 14x - 0.2x^2.
Please use an ^ before exponents.
The function has a maximum value where the derivative is zero.
f'(x) = 14 - 0.4 x = 0
x = 14/.4 = 35
If you don't know differential calculus yet, you can get the same result by "completing the square", but it takes longer, espcially with decimals involved.
Determine whether the system is consistent, inconsistent, or dependent.
3x + 2y = 15
6x + 4y = 30
inconsistent
To find the number of items the company should sell in order to maximize their profit, we need to find the maximum point of the profit function. The profit is given by the equation:
P(x) = f(x) - C(x)
Where:
P(x) is the profit function
f(x) is the revenue function
C(x) is the cost function
In this case, the cost function is not provided. However, we can assume a simplified cost function by using a constant cost per item sold. Let's assume the cost per item sold is c. Therefore, the cost function can be written as:
C(x) = cx
Substituting the revenue and cost functions into the profit function, we get:
P(x) = (14x - 0.2x^2) - (cx)
= 14x - 0.2x^2 - cx
To maximize the profit, we differentiate the profit function with respect to x and set it equal to zero:
dP(x)/dx = 14 - 0.4x - c = 0
Solving for x, we get:
0.4x = 14 - c
x = (14 - c)/0.4
To maximize the profit, the value of x should be positive. So, we need to find the minimum value of c such that (14 - c)/0.4 is positive.
Once we have determined the appropriate cost per item sold, we can find the corresponding number of items to be sold to maximize the profit using the formula x = (14 - c)/0.4.