Respuesta :
The amount spent on an advertisement that will result in maximum profit is $40,000 and that maximum profit is $5,450,000.
How do we calculate maximum profit from the profit function?
a) The maximum profit of the company occurs when the first derivative of the profit function is equal to zero as follows:
p’(x) = -10x + 400 = 0
b) The amount spent on an advertisement that will result in the maximum profit can be calculated from the maximum profit function as follows:
-10x + 400 = 0
x = -400 / -10
x = 40 in thousand dollars
Substituting x = 40 into the profit function, we have:
p(x) = (-5 * 40²) + (400 * 40) – 2550 = 5,450 in thousand dollars
Explanation: The calculation above implies that the amount spent on an advertisement that will result in the maximum profit is $40,000 and that the maximum profit is $5,450,000.
c) To calculate the amount that must be spent on advertising to obtain a profit of at least $54,000,000 (or 54,000 in thousand dollars), we equate it to the profit function and solve for x as follows:
54,000 = -5x²+400x-2550
54,000 + 2,550 = -5x²+400x
56,550 = -5x²+400x
5x² - 400x + 56,550 = 0 .................................. (1)
Using the almighty formula as follows:
x = (-b +/- (b^2 - 4ac)^0.5) / 2a ...................... (2)
Where, from equations (1) & (2), we have:
a = 5
b = -400
c = 56,550
Substituting the values into equation (2), we have:
x = (-(-400) +/- (-400^2 - (4 * 5 * 56,550))^0.5) / (2 * 5)
x = Undefined
Since x = undefined, this implies that there is no unique solution to this problem.
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