Consider the following problem: The total expected revenue from selling tickets for a certain concert as a function of a single ticket's price, x, changes at a rate of r(x)=17-0.24 x thousands of dollars per dollar. When x=70, the total expected revenue is 128 thousand dollars. What is the total expected revenue when the ticket price is $80 ? Which expression can we use to solve the problem? Choose 1 answer: (A) r^(')(80)-r^(')(70) (B) 128+int_(70)^(80)r(x)dx (C) int_(0)^(80)r(x)dx (D) r^(')(80)

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Consider the following problem: The total expected revenue from selling tickets for a certain concert as a function of a single ticket's price,  x, changes at a rate of  r(x)=17-0.24 x thousands of dollars per dollar. When  x=70, the total expected revenue is 128 thousand dollars. What is the total expected revenue when the ticket price is  $80 ? Which expression can we use to solve the problem? Choose 1 answer: (A)  r^(')(80)-r^(')(70) (B)  128+int_(70)^(80)r(x)dx (C)  int_(0)^(80)r(x)dx (D)  r^(')(80)

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Define Rate of Change Function: To find the total expected revenue when the ticket price is $80, we need to consider how the revenue changes as the ticket price changes from $70 to $80. The rate of change of the revenue with respect to the ticket price is given by the function r(x) = 17 - 0.24x. To calculate the change in revenue, we need to integrate the rate of change function r(x) from the initial ticket price to the final ticket price.Calculate Total Expected Revenue: The correct expression to calculate the total expected revenue when the ticket price increases from $70 to $80 is the integral of r(x) from 70 to 80, added to the initial revenue when x was $70. This is because the integral of the rate of change function gives us the total change in revenue over the interval from $70 to $80.Determine Integral Expression: The expression that represents this calculation is option (B) 128 + ∫ from 70 to 80 of r(x) dx. This is because we start with the initial revenue of 128 thousand dollars when x is $70, and we add the change in revenue as the ticket price increases to $80.Evaluate Antiderivative at Limits: Now we need to calculate the integral of r(x) from 70 to 80. The integral of r(x) = 17 - 0.24x with respect to x is the antiderivative of r(x), which is 17x - 0.12x^2 (since the antiderivative of 17 is 17x and the antiderivative of -0.24x is -0.12x^2).Calculate Change in Revenue: We evaluate the antiderivative at the upper and lower limits of the integral and subtract the lower limit value from the upper limit value. This gives us the change in revenue from when x is $70 to when x is $80.Evaluating the antiderivative at x = 80 gives us 17(80) - 0.12(80)^2 = 1360 - 0.12(6400) = 1360 - 768.Evaluating the antiderivative at x = 70 gives us 17(70) - 0.12(70)^2 = 1190 - 0.12(4900) = 1190 - 588.The change in revenue is the difference between these two values: (1360 - 768) - (1190 - 588) = 592 - 602 = -10 thousand dollars.

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