As you enter the spacecraft, you look to connect your computer. To your dismay, the controls are also password protected. Your computer helpfully translates the prompt and notifies you that three seperate numbers are needed. Your computer is able to determine the algorithm for determining each number, along with some password hints. The computer displays the below information: Password Hints: 7 f(x) dx = -3 7 g(x) dx 9(2) da=-8 -4 4 | 9(2) da= -4 -4 7 First Number: (-2 f(x)-2g(z)) dz. -4 -4 - 7 Second Number: (2 f(x) 2 g(x)) dx + f f (x) dx 7 Third Number: 2g(x)dx de 4 First Number 22 Second Number Third Number -22 -8 Explain, in your own words and with your own work, how you arrived at this result. Be sure to explain using calculus concepts to best support the work of the game design team. Plotting the Course With access to the spacecraft's controls, you connect your computer and begin installing the fuel tank and start capacitor. Your computer informs you that it can plot most of the course to your home planet but needs assistance with determining your velocity and vertical position at a critical moment in time. Your computer provides you with a function that models your acceleration during the first few seconds of flight. You will need to take this information and calculate your velocity and vertical position at time t = 2 s. Your computer notes that you can use the below initial conditions in supporting your calculations: Initial time: 0 s Initial vertical position: 0 m Initial velocity: 0 m/s Acceleration during this time: a (t) = 65+4 + 53t m/s What values (to 2 decimal places) do you provide your computer? v (2) = 522 m/s $ (2) = 209.33 m Explain, in your own words and with your own work, how you arrived at this result. Be sure to explain using calculus concepts to best support the work of the game design team. Connecting the Start Capacitor The spacecraft is nearly ready for takeoff! The last task is to attach the start capacitor to the ship in a simple RC-circuit. Your computer warns you that some adjustments may need to be made if the heat generated in the resistor over the initial time interval is higher than a given threshold. Your computer is able to provide the following information, along with a note that the units are already accounted for in the equations: V.2 4 b Heat Generated in Resistor over the first 4 seconds: Q = R -2t/(RC) dt joules Voltage of the Battery: V = 2 volts Resistance: R = 2 ohms Capacitance: C = farads Maximum Heat Allowed in Resistor: Qmax=1 joules Is the heat generated in the resistor over the first 4 seconds below the maximum allowable threshold? Yes No What is the difference between the maximum threshold (provided above by your computer) and the amount of heat generated in the resistor (that you calculated)? Provide a positive value up to three decimal places. Number joules Explain, in your own words and with your own work, how you arrived at this result. Be sure to explain using calculus concepts to best support the work of the game design team.