Math 1300 Written Homework #10 Solutions 4.2, Q. 24. Let y = at 2e ...

a. Let g(v) be the fuel consumption of the same aircraft, but measured in gallons per mile
instead of gallons per hour. What is the relationship between f(v) and g(v)?
Solution: Suppose v = 100 miles per hour; then from the graph, f(100) = 50 gallons
per hour. So in the next one hour, the aircraft will travel about 100 miles and use up
about 50 gallons of fuel. Assuming the speed and fuel consumption stay constant over
that hour, the plane is using 50 1 = gallons of fuel per mile traveled.
100 2
More generally,
g(v) = = f(v) gallons
v mile .
f(v) gallons
hour
v miles
hour
Notice how the units “gallons per hour” divided by the units “miles per hour” cancels out
to give “gallons per mile,” which are the units we were supposed to get. Units confirm
that we are thinking about the problem in the right way.
b. For what value of v is f(v) minimized?
Solution: We just want the critical point, where f ′ (v) = 0. This seems to occur around
v = 220 miles per hour.
c. For what value of v is g(v) minimized?
Solution: g(v) will have the same general shape as f(v), with a vertical asymptote at
v = 0 and increasing toward infinity as v goes to infinity. So we just need the critical
point of g(v).
Using the Quotient Rule, we get
g ′ (v) = d
dv
f(v)
v
= vf ′ (v) − f(v)
v2 .
The critical point we care about is wherever f(v) = vf ′ (v). The best way to see this
from the graph is to notice that geometrically, we will have
f(v)
v = f ′ (v)
2