Part I
The change in annual revenue in thousands of dollars for the Ozark Zip Line Corporation has been modeled by the function.
R(x) = 8.2x^2 + 23.1x
where x is the number of years after 2000. To find the total revenue gained between the years 2002 and 2005, start by choosing a constant (C)530
Part II
Build the definite integral that is used to find the total revenue gained between the years 2002 and 2005
Part III
Apply the power rule to integrate R(x) over the given interval, and round your answer to the nearest whole tens of dollars.
Part I:
To find the total revenue gained between the years 2002 and 2005, we need to calculate the definite integral of the revenue function R(x) over the interval from 2002 to 2005.
Part II:
To build the definite integral, we need to define the limits of the interval. The interval from 2002 to 2005 can be represented as x = 2 to x = 5, where x is the number of years after 2000.
The definite integral for the total revenue gained between 2002 and 2005 can be written as:
∫[2 to 5] (8.2x^2 + 23.1x) dx
The integral sign (∫) represents the operation of integration. The function inside the integral represents the revenue function R(x), which is 8.2x^2 + 23.1x. The limits of integration [2 to 5] define the interval from 2002 to 2005.
Part III:
To solve the integral, we can apply the power rule of integration, which states that the integral of x^n is (1/(n+1))x^(n+1).
Using the power rule, we can integrate each term of the revenue function separately:
∫(8.2x^2 + 23.1x) dx
= (8.2/3)x^3 + (23.1/2)x^2 + C
Now we can substitute the limits of integration and subtract the value of the integral at the lower limit from the value of the integral at the upper limit to find the total revenue gained between 2002 and 2005:
Total revenue = [(8.2/3)(5^3) + (23.1/2)(5^2)] - [(8.2/3)(2^3) + (23.1/2)(2^2)]
(=436 - 47.2)
Rounding to the nearest whole tens of dollars, the total revenue gained between 2002 and 2005 is $390.
Please note that the constant (C) mentioned in Part I is not used in the calculation of the definite integral or the total revenue gained.