Optimization prob.: Find most economical speed of boat

[sickplaya]

A man rents a large boat to make a 400 km trip. He has to pay at the rate of $15 per hour for the boat and he must pay for the gasoline at $1 per gallon. When the boat is travelling at x km per hour, it burns gasoline at the rate of 10 + x^2/16 gallons per hour. What is the most economical speed?

so t = 400/x
Let C(x) = cost function
C(x) = 15(400/x) + (400/x)(10 + x^2/16)(1)
C(x) = 10000x^-1 + 25x
C'(x)= -10000x^-2 + 25=0
so x= 20 km/h

im wondering if 20km/h is right and if someone could verify my steps that would be very appreciated.
thanks.

 

[arthur ohlsten]

Distance D=400km
time of trip = t hours
gasoline costs $1 gallon
gallons used = [10+x^2/16]t
x= boats speed[km/ hr]

t=400/x

cost =15t + [10+x^2/16]t
t cost = 15t^2 +[10+x^2/16]t^2
[xt]^2=400^2
t cost = 15t^2+10t^2+160000/16
cost = 25t + 10000t^-1 take derivative
d[cost]/dt = 25 -10,000t^-2
set d[cost]/dt =0
0= 25 -10,000/t^2
10,000/t^2=25
t^2=400
t=+/- 20hr
t must be positive
t=20 hr
400/20= 20 km/hr answer
Arthur
t=+/- 1/20 hr time can't be negative
t=1/20 hr

 

[soroban]

Hello, sickplaya!

A man rents a large boat to make a 400 km trip.
He has to pay at the rate of $15 per hour for the boat
. . and he must pay for the gasoline at $1 per gallon.
When the boat is travelling at \(\displaystyle x\) km/hr,
. . it burns gasoline at the rate of \(\displaystyle 10\,+\,\frac{x^2}{16}\) gallons/hr.
What is the most economical speed?

So: \(\displaystyle \,t\:=\:\frac{400}{x}\)

Let \(\displaystyle C(x)\) = cost function

\(\displaystyle C(x) \:= \:15\left(\frac{400}{x}\right)\,+\,\left(\frac{400}{x}\right)\left(10\,+\,\frac{x^2}{16}\right)\cdot(1)\;\;\Rightarrow\;\;C(x) \:=\: 10000x^{-1}\,+\,25x\)

\(\displaystyle C'(x) \:=\:-10000x^{-2}\,+\,25\:=\:0\;\;\Rightarrow\;\;x\:=\:20\text{ km/hr}\)

Absolutely right! . . . Lovely work!


Your answer is obviously for minimum cost
. . since there doesn't seem to be a maximum cost.

We can, of course, verify this with the Second Derivative Test,
. . but I have a baby-talk approach.

He could drive at a very slow speed (inches per day).
But since the rental is $15/hr and gasline costs at least $10 per hour
. . and the trip might take months . . . forget it!

He could rent a hypersonic boat and drive at, say, Mach 2.
. . The trip would be quite short; he'd save on the rental fee.
But at thousands of kilometers per hour,
. . the cost of gasoline would be ridiculously large.