A fast-food restaurant determines the cost and revenue models for its hamburgers

Question

A fast-food restaurant determines the cost and revenue models for its hamburgers. C = 0.3x + 7300, 0 lessthanorequalto x lessthanorequalto 50,000 Write the profit function for this situation. Determine the intervals on which the profit function is increasing and decreasing. (Enter your answers using interval notation.) increasing decreasing Determine how many hamburgers the restaurant needs to sell to obtain a maximum profit. hamburgers Explain your reasoning. Because the function changes from decreasing to increasing at this value of x, the maximum profit occurs at this value. The restaurant makes the same amount of money no matter how many hamburgers are sold. Because the function changes from increasing to decreasing at this value of x, the maximum profit occurs at this value. Because the function is always decreasing, the maximum profit occurs at this value of x. Because the function is always increasing, the maximum profit occurs at this value of x.

Explanation / Answer

a)

p(x) = R(x) - C(x)

= (1/10000)*(65000 x - x^2) - ( 0.3*x + 7300)

= 6.5x - 0.0001*x^2 - 0.3*x - 7300

= -0.0001*x^2 + 6.2*x - 7300

b)

p(x) = -0.0001*x^2 + 6.2*x - 7300

p'(x) = -0.0002*x + 6.2

p'(x) = 0

-0.0002*x + 6.2 = 0

x = 31000

for x<31000 ---> p'(x) is positive

for x>310000 ---> p'(x) is negative

So,

increasing : [0,31000)

decreasing: (31000,50000)

c)

for maximum profit,

number of hamburgers = 31000

because the function changes from increasing to decreasing at this value of x, the maximum profit occur at this value

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