SOLUTION: With a 12 gallon tank, a Jupiter gets 22 mi/gal. Engineers estimate that every 2 gallon increase in tank size causes gas mileage to decrease by 1 mi/gal. What should the size of

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Question 180862: With a 12 gallon tank, a Jupiter gets 22 mi/gal. Engineers estimate that every 2 gallon increase in tank size causes gas mileage to decrease by 1 mi/gal. What should the size of the tank be for the Jupiter to have the greatest range (number of miles on a tank of gas)?

Found 2 solutions by kev82, ankor@dixie-net.com:
Answer by kev82(151) (Show Source):
You can put this solution on YOUR website!
v = volume of tank
r = miles per gallon
distance = vr
dr/dv = -0.5 (every increase of 2 decreases by 1)
r = c - 0.5v (integrate)
22 = c - 6 (substitute r=22,v=12)
c = 28
v(28-0.5v) = distance (substitute)
28-0.5v -0.5v = 28-v = 0 (differentiate and solve for turning point)
v = 28
r = 28 - 14 = 14 (calculate r at turning point)
14*28 = 392 (calculate distance)

Answer by ankor@dixie-net.com(22740) (Show Source):
You can put this solution on YOUR website!
Here's another way to do this:
:
With a 12 gallon tank, a Jupiter gets 22 mi/gal. Engineers estimate that every 2
gallon increase in tank size causes gas mileage to decrease by 1 mi/gal.
What should the size of the tank be for the Jupiter to have the greatest range
(number of miles on a tank of gas)?
:
Let x = increase in size (in gal) of the tank
then
y = range of the tank (in miles)
:
It says, "every 2 gallon increase in tank size causes gas mileage to decrease by 1 mi/gal."
From this we can say it decreases .5 mi mpg, for each 1 gal increase (.5x)
:
Range = mpg * gal
y = (22-.5x)(12+x)
FOIL
y = 264 + 22x - 6x - .5x^2
:
y = -.5x^2 + 16x + 264; a quadratic equation
:
Max range occurs at the axis of symmetry, can be found using x =
In this equation: a = -.5, b = 16
x =
x =
x = +16 gal increase for max range
:
Therefore: 12 + 16 = 28 gal tank will give max range
:
Actual range: (22 - .5(16)) * 28 = 392 mi