Respuesta :
Answer:
The maximum profit that can make by the company is $461.
Step-by-step explanation:
Given expression
[tex]y=-2x^2+81x-359[/tex]
y = the amount of profit
x= Number of widget.
We know that,
The maximum value of a function [tex]y=ax^2+bx+c[/tex] only when the value of [tex]x=-\frac{b}{2a}[/tex]
Here a= -2 and b=81
The profit of the company will maximum only when if they sell [tex]x=-\frac{81}{2.(-2)}\approx 20[/tex] widget.
To find the maximum profit putting x= 20 in given expression.
[tex]y=-2(20)^2+81.20-359[/tex]
=$461
The maximum profit that can make by the company is $461.
The shape of the curve of the graph of the function of the company's profit
is a concave down parabola.
The maximum profit the company can make is approximately $461
Reasons:
The function for the profit the company makes, y = -2·x² + 81·x - 359
Where;
x = Selling price of each widget
Required:
The companies maximum profit given to the nearest dollar
Solution:
The value of the maximum profit is given by the maximum value of the
function, [tex]y_{max}[/tex], the x-value of which is given when we have;
[tex]\dfrac{dy}{dx} = 0[/tex]
At the maximum value (profit), we have;
[tex]\dfrac{dy}{dx} = \dfrac{d}{dx} \left(-2 \cdot x^2 + 81 \cdot x - 359 \right ) = -4 \cdot x + 81 = 0[/tex]
Therefore;
-4·x + 81 = 0
[tex]x = \dfrac{81}{4} = 20.25[/tex]
At the point of maximum profit, x = 20.25, therefore;
The maximum profit, [tex]y_{max}[/tex] = -2×20.25² + 81×20.25 - 359 = 461.125
The maximum profit to the nearest dollar, [tex]y_{max}[/tex] = $461
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