Max/Min Problem using first and second derivatives?

Question

Using first and second derivatives, solve this question:

A truck is to be driven 300 miles on a freeway at a constant speed of X mph. Speed laws require X to be greater (or equal) to 30 and less than (or equal to) 60 MPH [so speed needs to be 30-60 mph].

Assume that the fuel cost 60 cents per gallon (yea I wish) and it is consumed at the rate of:

2 + (x^2)/600 gallons per hour. If the driver's wages are 8 dollars per hour and if he obeys all speed laws, find the most economical speed.

If you can please show your work when solving this, it will be much appreciated =] THE ANSWER IS 60 MPH, but I don't know how to get there.

Thanks =]

Answer 1

let G(x) be eq. for gas/fuel and W(x) be eq. for wage, then cost would be C(x) = G(x) + W(x).

30 <= x <= 60

G(x) = 300*0.6*[2 + (x^2)/600] / x
= 360/x + 0.3x

W(x) = 8*300/x
= 2400/x

C(x) = G(x) + W(x)
= 0.3x + 2760/x

C'(x) = 0.3 - 2760/x^2 = 0
x^2 = 2760/0.3
= 9200
x = 20√23 = 95.92 mph

since the highest speed allowed is 60, so that'd be the answer.

Answer 2

Hi.

If x is the speed driven, then it will take 300/x hours to drive the whole way.

If gas costs 60 cents a gallon and is consumed at the rate of 2 + (x^2)/600 gallons per hour, then the cost of the gas for 300/x hours is found from:

.60[ 2 + (x^2)/600](300/x)

In addition to the expense of the gas, the driver must be paid $8 an hour for 300/x hours, which is 8*300/x or 2400/x

The total cost of the trip is found from the equation:

Y = .60[ 2 + (x^2)/600](300/x) + 2400/x, where 30 ≤ x ≤ 60.

If you graph this on a graphing calculator, the minimum value occurs when x = 60. At 60 mph, the gas would cost .60(2 +60^2/600)(300/60) or .60(8)(5) = $24. Paying the driver for 300/60 or 5 hours would cost 5 x 8 or $40. This gives a total of $64 for the trip.

I hope that helps!! :-)

Answer 3

I graphed 2 and the minimum turn into at (0,0) staring on the graph there do not seem any inflection factors because of the fact the slope is increasing over the whole graph (different than the place this is discontinuous) And this additionally ability it is not in any respect concave down