Calculus\Word Problem\Calculus Word Problem Help

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ROAD_DOGG33J

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Here is the problem:

For speeds between 40 and 65 mph (miles per hour), a truck gets 480/x mpg (miles per gallon)
at a constant speed of x mph (miles per hour). Diesel gasoline costs $2.23 per gallon and the
driver is paid $15 per hour. What is the most economical constant speed between 40 and
65 mph at which to drive the truck?
Click to expand...

I figure that by most economical they mean the lowest cost, so cost is equal to cost of diesel + cost of the driver($15 per hour). I'm not really sure how to go about setting it up.
 
I think I did something wrong, but my brain is a bit fried from studying for finals.

C_total/hour = C_driver/hour + C_gas/hour

C_driver/hour = $15/hour

C_gas/hour = ($2.23 / 1 gallon) * (x gallon / 480 miles) * (x miles / 1 hour)

= $2.23*x*x/480 hour

=> C_total = 15 + 2.23x^2/480

If C_total is correct, then obviously it’s an upward-opening parabola, in which case the minimum will be 40.
 
Ah... wouldn't you not care how long it takes... but how far you get?

Ergo, how much does a (say) 100 miles journey cost, at 40, 65 or anything inbetween. The nature of the question suggests there is a speed somewhere between 40 and 65 which it is cheapest to cover a mile in.
 
I don't think that's correct Sage
Driving 60mph can get you 20miles further every hour than driving 40mph

total cost for 100miles = 100/x * 15 + 100/(480/x) * 2.23
x = speed
total must be minimum

Edit: i made a mistake, fixed it
 
DustDriver
total cost = x * 15 + 480/x * 2.23
Click to expand...
Simple unit analysis shows that this is an impossible equation.

RHS: miles*dollars + miles*dollars/gallon

You’re forgetting that as a vehicle goes faster, mpg goes down. Otherwise, we’d all be saving money by driving our cars as fast as possible.
 
Venari
The nature of the question suggests there is a speed somewhere between 40 and 65 which it is cheapest to cover a mile in.
Click to expand...

I think all or almost all questions we covered so far had the min/max at a critical point. I guess this is one of those questions where you have to look at the endpoints.


Thanks for all the help so far
 
Heh, yeah, was about to point that out, DD.

The answer is about 57 mph, at just under $0.528/mile.

cost/mile = 15/x + 2.23x/480

6d5620eb.jpg
 
Accidentally forgot to switch around 480/x to x/480 when i removed the 100 out off my formula :dopey:
 
Nope.

x is in miles / hour. You want miles as the denominator, so it is 15 $/h divided by x miles/h (or 15 $/h * 1/x h/mile - and the hours cancel, leaving you 15/x $/mile )
 
Yes, Venari is correct – my mistake was in calculating cost/hour instead of cost/mile. In retrospect, that was a pretty stupid thing to do. :dunce:
 

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