investigate a particular geothermal vent or not. A useful quantity for our purposes is the reproductive distance rate F of a geothermal vent. This measures how much the extremophiles reproduce per metre from the centre of the geothermal vent and has units of 104 bacteria per metre times seconds. For a generic vent, the distance rate F has been measured to give the following table. I--Inmmll I----Inmm {a} {2 points) Write down {out do not evaluate!) an integral expression for the rate R at which extremophiles reproduce a given distance 49' away from the centre of the geothermal vent provided you know the rate of reproduction is 0 at the centre of the geothermal vent. {Hint R has units 104 bacteriais) (Hint: if F was constant, then R : F X J) {b} {3 points} Using right-hand Riemann sums, approximate R as a function of ii. That is, fill out the following table I-----Inllni ||---------| {c} (2 points} Extremophiles can only exist a particular distance away from a geothermal vent depending on the temperature of the vent. We can model this by assuming that all extremophiles exist a distance a = 4.0041"? + 3.41" 719 from the centre of the vent, where T is the temperature of the vent in units of "C and varies between 400 C and 450 (3. Modify your expression in (a) to give the rate of reproduction R as a function of T. {d] {3 points] Given that it's quite easy to measure the temperature of a vent remotely, we can use part [c] to find our ideal vent to send missions. Find the temperature T which maximizes the rate of reproduction of extremophiles