Question

The annual profit for a company that manufactures cell phine accessories can be modeled by the function P(x)=-0.0001x^2+70x+12500 where x is the numbe

Functions
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asked 2021-05-09
The annual profit for a company that manufactures cell phine accessories can be modeled by the function
\(\displaystyle{P}{\left({x}\right)}=-{0.0001}{x}^{{2}}+{70}{x}+{12500}\) where x is the number of units sold and P is the total profit in dollars.
a. What sales level maximizes the company's annual profit?
b. Find the maximum annual profit for the company.

Expert Answers (1)

2021-05-10

a.The sales level needed to maximize the annual profit is the x-value of the vertex of the quadratic function. The highest point is the vertex of the parabola represented by the equation in the form y=ax2+bx+c with a<0 where the x-coordinate is given by x=−b2a.
Substitute a=−0.0001 and b=70:
\(\displaystyle{x}=-{\left(\frac{{70}}{{{2}{\left(-{0.0001}\right)}}}\right)}={350000}\)
So, 350,000 units will maximize the profit.
b.Using the x-value from part (a) and the profit function, the maximum annual profit is:
\(\displaystyle{P}{\left({350000}\right)}=-{0.0001}{\left({350000}\right)}^{{2}}+{70}{\left({350000}\right)}+{125000}\)
\(P(350000)=12262500\)
So, the maximum profit is $12,262,500.

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