In 2008, the price of gasoline is the Unites States spiked and then dropped. The
ID: 2850356 • Letter: I
Question
In 2008, the price of gasoline is the Unites States spiked and then dropped. The average monthly price (in cents per gallon) of unleaded regular gasoline for 2008 can be approximated by the function p(t) = 0.614t^3 + 6.25t^2 + 1.94t + 297, for 0 < t < 25, where t is in months and t = 1 corresponds to January 2008. (a) Use calculus to determine the intervals on which the price is increasing. (b) Determine the intervals on which the price is decreasing. (c) Find any relative extrema for the price of gasoline, as well as when they occurred.
Thank you.
Explanation / Answer
p(t) = 0.614t3 + 6.25t2 + 1.94t + 297
p '(t) = -0.614(3)t2 + 6.25(2)t + 1.94
==> p '(t) = -1.842t2 + 12.5t + 1.94
critical points ==> p '(t) = 0
==> -1.842t2 + 12.5t + 1.94 = 0
==> t = [-12.5 + (12.52 -4(-1.842)(1.94))]/2(-1.842) , [-12.5 - (12.52 -4(-1.842)(1.94))]/2(-1.842)
==> t = -0.1518 , 6.93791
t cannot be negative hence t = 6.93791
a) price increasing ==> p '(t) > 0
==> -1.842t2 + 12.5t + 1.94 > 0
==> t belongs to (0 , 6.93791)
b) price decreasing ==> p '(t) < 0
==> -1.842t2 + 12.5t + 1.94 < 0
==> t belongs to (6.93791, 25)
c) p '(t) = -1.842t2 + 12.5t + 1.94
==> p ''(t) = -1.842(2t) + 12.5
==> p ''(t) = -3.684t + 12.5
p ''(6.93791) = -3.684(6.93791) + 12.5 < 0
Hence at t = 6.93791 price is maximum
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