MTH 178 CCC Maximize the Profit Question

I’m working on a calculus project and need the explanation and answer to help me learn. we came up with a word problem and are trying to use the trolley system and find the maximum profit using marginal revenue, marginal cost, and derivatives. To maximize the profit from public transportation systems the concept of marginal revenue and marginal cost can be used. By setting the marginal revenue equal to the marginal cost and then determining the level of output that corresponds to this point. Minimum and maximum derivatives can also be used to optimize systems and achieve desired outcomes as it helps identify the highest point of profit through the slope curve. To add on, using the minimum derivative optimizes public transportation systems by identifying where the point of the slope cost is at its lowest. Derivatives of a function are used to calculate speed or distance traveled, such as miles per hour and kilometers per hour. This is a measurement that is used to determine how quickly the derivative changes the value of Y with the change in the value of X. Derivatives are used every day by certain employers to help determine something changing. For example, the change of speed in trains, buses, cars, and planes is solved with calculus.The formulas we plan on using is derivative of a function: F(x) = lim (h->0) [f(x)(x+h) – f(x))/h]This formula gives the slope of the tangent line to the graph of the function f(x) at any given point x. The derivative is forming a base for a plan in calculus and is used to find the rate of change of a function. To find the marginal cost and marginal revenue, we take the derivative of the cost function and revenue function. The marginal cost is the derivative of the total cost function with respect to the quantity of output: MC(q) = dC(q)/dqMC(q) is the marginal cost, C(q) is the total cost function, and q is the quantity of output. MR(q) = dR(q)/dqMR(q) is the marginal revenue, R(q) is the total revenue function, and q is the quantity of output.By incorporating these factors in calculus we can come up with the speed the trolley must travel within a certain time frame so they can obtain the maximum profit.The San Diego MTS travels at 55 mph while covering 65 miles of San Diego. Its annual income rate was 360.0 million in 2021. While the day pass is $6.00 and the monthly pass is $72.00, charging citizens $0.50 per mile the ride will increase the annual income towards The San Diego MTS. According to sdmts.com, an average rider travels 15 miles a day. Changing charges from $6.00 a day to $0.50 per mile will increase the income revenue since an average person travels at least 15 miles a day. Considering an individual has to ride at least 15 miles to work and another 15 miles back the MTS is earning more per mile than they would for a day pass. Jack uses the trolley every Monday, Wednesday, and Thursday for work, 3 days a week. He travels 28 miles a day from Santee to Mission Valley and back to Santee. How much is he spending per week with the new prices of $.50 per mile compared to the $ 6.00-day pass?