A useful quantity for our purposes is the reproductive distance rate F of a geothermal vent....

  

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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. 1 1.5 2 2.5 3 3.5 4 0.2 -0.3 -0.8 -1.2 distance from vent (m) 0 0.5 F (10 bacteria/ms) 0 0.5 0.78 1 1.2 (a) (2 points) Write down (but do not evaluate!) an integral expression for the rate R at which extremophiles reproduce a given distance d 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 bacteria/s) (Hint: if I was constant, then R = Fd) (b) (3 points) Using right-hand Riemann sums, approximate R as a function of d. That is, fill out the following table d (m) R (10' bacteria/s) 0 0.5 1 1.5 2 2.5 3 3.5 4 (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 d = -0.00472 + 3.4T - 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 C. 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. 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. 1 1.5 2 2.5 3 3.5 4 0.2 -0.3 -0.8 -1.2 distance from vent (m) 0 0.5 F (10 bacteria/ms) 0 0.5 0.78 1 1.2 (a) (2 points) Write down (but do not evaluate!) an integral expression for the rate R at which extremophiles reproduce a given distance d 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 bacteria/s) (Hint: if I was constant, then R = Fd) (b) (3 points) Using right-hand Riemann sums, approximate R as a function of d. That is, fill out the following table d (m) R (10' bacteria/s) 0 0.5 1 1.5 2 2.5 3 3.5 4 (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 d = -0.00472 + 3.4T - 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 C. 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.

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