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A gasoline refinery found that the cost to produce 12,50012,500 gallons of gasoline last month was $27,000$27,000
A gasoline refinery found that the cost to produce 12,50012,500 gallons of gasoline last month was $27,000$27,000. At that time, the cost was going up at a rate of $1200$1200 per month, while the number of gallons of gasoline the refinery produced was going up at a rate of 350350 gallons per month.
At what rate was the average cost to produce a gallon of gasoline increasing or decreasing last month?
Expert Solution
We may assume the cost as C(t)C(t) a function of time and the quantity as q(t)q(t) a function of time.
The average cost function formula is ¯C(t)=C(t)q(t)C¯(t)=C(t)q(t).
Let us assume t=tlt=tl and it is considered as last month.
Given, C(t)=$ 27,000C(t)=$ 27,000, q(t)=12,500q(t)=12,500, C′(t)=$ 1200C′(t)=$ 1200 and q′(t)=350q′(t)=350.
Calculating the marginal average cost:
¯C′(tl)=q(tl)C′(tl)−C(tl)q′(tl)(q(tl))2=(12500)(1200)−(27000)(350)(12500)2=5550000(12500)2=1113125¯C′(tl)=0.03552C¯′(tl)=q(tl)C′(tl)−C(tl)q′(tl)(q(tl))2=(12500)(1200)−(27000)(350)(12500)2=5550000(12500)2=1113125C¯′(tl)=0.03552
Hence, the average cost is increasing at rate of $ 0.03552$ 0.03552 per gallon for per month .
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