Re: word problem and i don't know how to even start, please
Hello, bsalerno!
This is an intricate problem, but they make us take baby-steps.
Suppose that the cost of operating a truck in Mexico is \(\displaystyle 45\,+\,0.29v\) cents/mile
when the truck runs at a steady speed of \(\displaystyle v\) mph. \(\displaystyle \;\)The top speed of the truck is 100 mph.
Assume that the driver is paid $8/hour to drive the truck, and he is to begin a 2600-mile trip.
(a) Write the cost of operating the truck in dollars, as a function of the speed \(\displaystyle v\), for the planned trip .
(b) Write the cost of driver's wages in dollars, as a function of the speed \(\displaystyle v\), for the planned trip .
(c) The total cost of the planned trip, as a function of the speed \(\displaystyle v\), is the sum of the first two costs.
Find the most economic speed for the planned trip, i.e., the speed that minimize the total cost.
(a) The truck will travel \(\displaystyle 2600\) miles at \(\displaystyle \,45\,+\,0.29v\) cents/mile.
\(\displaystyle \;\;\;\)The operating cost is: \(\displaystyle \,2600(45\,+\,0.29v)\)
cents \(\displaystyle \:=\:26(45\,+\,0.29v)\)
dollars.
(b) The driver goes \(\displaystyle 2600\) miles at [texx]v[/tex] mph.
\(\displaystyle \;\;\;\)It will take him: \(\displaystyle \frac{2600}{v}\) hours.
\(\displaystyle \;\;\;\)At $8\hour, he gets: \(\displaystyle \,8\,\times\,\frac{2600}{v}\:=\:\frac{20,800}{v}\) dollars.
(c) The total cost is: \(\displaystyle \,C\;=\;26(45\,+\,0.29v)\,+\,20,800v^{-1}\;=\;1170\,+\,7.54v\,+\,20,800v^{-1}\)
Differentiate and equate to zero: \(\displaystyle \:C'\:=\:7.54\,-\,20,800v^{-2}\:=\:0\)
Multiply by \(\displaystyle v^2:\)
\(\displaystyle \;\;\;7.54v^2\,-\,20,800\:=\:0\;\;\Rightarrow\;\;7.54v^2\,=\,20,800\;\;\Rightarrow\;\;v^2\,=\,\frac{20,800}{7.54}\,=\,2758.62069\)
Therefore: \(\displaystyle \,v\:=\:\sqrt{2758.62069}\:=\:52.52257314\:\approx\:\)
52.5 mph
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Ha! . . . I'm trying to visualize this driver.
From a full stop, he drives at exactly 52.5 mph ... no acceleration allowed!
Then he drives for over 49 hours (2<sup>+</sup> days) at
exactly that speed,
\(\displaystyle \;\;\)no stopping, no slowing (even for toll booths, traffic, and towns).
Sounds like a script for "Speed 3".
