a)
y=C₁(v)=Fuel cost for the trip as a linear function of the average speed, v.
x=v= average speed.
Data:
When: v=60 mph ⇒ C(v)=$30
When: v=120 mph ⇒ C(v)=$42
We have two points (60,30) and (120,42), therefore we have to calculate the equation of the line passes through these points.
1) We calculate the slope of the function:
Given two points (x₁,y₁) and (x₂,y₂) the slope of the line passes through these points will be:
m=(y₂-y₁)/(x₂-x₁)
In our case, the points are (60,30) and (120,42) and the slope will be:
m=(42-30)/(120-60)=12/60=0.2 ($0.2 per mile)
slope-intercept form of a line: we need a point (x₀,y₀) and the slope:
y-y₀=m(x-x₀)
We know the slope (m=0.2) and we can choose either of the points known, the result is always the same with any of these points known. Therefore:
(60,30)
m=0.2
y-y₀=m(x-x₀)
y-30=0.2(x-60)
y-30=0.2x-12
y=0.2x-12+30
y=0.2x+18
y=C(v)
x=v
C₁(v)=0.2v+18
Answer (a)= the fuel cost ( C(v) ) for the trip as a linear function of the average speed, v would be: C₁(v)=0.2v+18
b)
y=C₂(v)=no- fuel cost of the trip en function of the average speed (v).
v=average speed
time=distance / speed
C₂(v)=$20 per hour (distance /speed)
C₂(V)=$20 / hour(100 miles / speed)
C₂(v)=2000/ v
For example:
if v=60 mph ⇒ C₂(v)=2000 /60≈$33.33
if v=120 mph⇒C₂(v)=2000/120≈$16.67
Other way
We have to calculate the time spend with the two speeds known.
if average speed=60 mph ⇒time=100 miles/(60 miles/hour)=5/3 hour.
if average speed=120 mph ⇒time=100 miles/(120 miles/hour)=5/6 hour.
non-fuel cost if v=60 mph =$20 per hour(5/3 hour)≈$33.33
non-fuel cost if v=60 mph =$20 per hour(5/6)≈$16.67
Answer (b): the function of the non-fuel cost for the trip will be :
C₂(v)=2000/v. The non-fuel cost for the speeds given are:
non-fuel cost if v=60 mph ≈$33.33
non-fuel cost if v=120 mph ≈$16.67
c)
C(v)=total cost for the trip
C(v)=C₁(v)+C₂(v)
C(v)=(0.2v+18)+(2000/v)
or
C(v)=(0.2v²+18v+2000)/v
1) We have to do the first derivative of this function:
C´(v)=0.2-2000/v²
C´(v)=(0.2v²-2000)/v²
2) We have to find the values of "v" when C´(v)=0
Then:
0.2v²-2000=0
v=√(2000/0.2)=100
3)We have to do the second derivative:
C´´(v)=4000/v³
C´´(100)>0 ⇒ we have a minimum at v=100
The total cost of the trip will be:
C(100)=0.2(100)+18+2000/100=$58
Answer (c): the average seed that minimizes the total cost of the trip will be 100 mph.
