Answer:
a. profit: P = -0.00005x² +2.65x -7500
b. increasing: x < 26500; decreasing: x > 26500
c. maximum profit: 26,500 hamburgers
Explanation:
Given cost and revenue functions C = 0.6x +7500 and R = (65000x -x²)/20000, you want the profit function, the intervals of increasing and decreasing profit, and the sales (x) required for maximum profit.
a. Profit function
The profit is the difference between revenue and cost.
P = R -C
P = (65000x -x²)/20000 - (0.6x +7500) . . . . . 0 ≤ x ≤ 50,000
P = -0.00005x² +2.65x -7500
b. Increasing/decreasing
The vertex of the profit function ax²+bx+c is found at x = -b/(2a). For this profit function, that is ...
x = -(2.65)/(2(-0.00005)) = 26,500
Since the leading coefficient is negative, we know the function is increasing for x values below the vertex value, and is decreasing for greater x-values.
Increasing: x < 26500
Decreasing: x > 26500
c. Maximum
The function is a maximum at the point where it changes from increasing to decreasing.
26500 hamburgers sold will give maximum profit.
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Additional comment
The graph shows the profit function. 3000 hamburgers must be sold to break even. Profit per hamburger is a maximum of 2.35 at that point, declining by 0.01 per hamburger for each 100 hamburgers sold more than that.
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