operates 16 oil wells in a particular ch additional well they add will reduce th

Question

operates 16 oil wells in a particular ch additional well they add will reduce the perwell ut e 2 field. Each well can extract 240 barrels of oil per day, well output by 8 barrels. They naturally average. wrant to maximize their total production. What is the primary equation? What do your variables represent? relationship among thoee variables. Remermber, this will probably be where the numbers given above get used. (c) Write the primary equation in terms of a single variable and fnd the appropriste critical valuns (d) Show that this critical value maximizes their production. How many additional wells should be added (or how many removed) to maximize their output?

Explanation / Answer

8)

(a)

let the total production be z, number of wells=x , average production per each well =y

primary equation:Total production = number of wells* average production per each well

=>z= xy

(b)

let the change in number of wells from the existing ones be a

so the change in average production per each well =-8a

secondary serationhip:number of wells ,x= 16+a, average production per each well ,y=240-8a

(c)

equation: z=(16+a)(240-8a) ,0<=a<=30

=>z=3840+112a-8a2

=>dz/da=0+112-16a

=>dz/da=112-16a

=>d2z/da2=0-16*1

=>d2z/da2=-16

for critical point, dz/da=112-16a=0

=>112-16a=0

=>a=112/16

=>a=7

at a=7, z=4232 (critical value)

(a)

at a=7, z=4232,d2z/da2=-16<0

at a=0, z=3840

at a=30,z=0

so critical value maximises production

7 additional wells should be added

maximum production output is 4232 barrels of oil per day

PLEASE RATE IF HELPFUL.PLEASE COMMENT IF YOU HAVE ANY DOUBT

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