400 3. A truck gets miles per gallon when driven at a constant speed of x miles

Question

400 3. A truck gets miles per gallon when driven at a constant speed of x miles per hour (between 50 and 80 miles per hour). The price of fuel is $3.54 per gallon and the driver gets paid $34 per hour. The truck is going to take a 2000 mile trip. If the driver drives at 50 miles per hour, how many miles per gallon will the truck get? Ar that mileage, how many gallons will the driver need for 2000 miles? How much will gas cost? a. How many hours will it take to drive 2000 miles at 50 miles per hour? (Ignore resting) How much will the driver be paid? b. c. What will be the total cost, gas plus driver, if the speed is 50 miles per hour? d.e.f Do parts a, b, and c again, but this time using a speed of 80 miles per hour. Now write a function C(x) that expresses total cost of the trip, as a function of x miles per hour. h. Find the derivative of your function, and use it to answer the next question. At what speed is it most economical to drive? (round to the nearest whole mile per hour) How do your answers to part c and part f help you determine that your answer will give minimum cost and not the maximum cost? i. At that speed, how much does the trip cost? (round to the nearest dollar)

Explanation / Answer

Solved the first 6 parts since similar (we need to solve first four parts as per Chegg guidelines), post one more question to get the answer to remaining four parts

a)

Miles per gallon at speed of 50 mi/hr = 400/50 miles per gallon = 8 miles per gallon

Number of gallons needed to cover 2000 miles = 2000/8 = 225 gallons

Cost of the gas = Number of gallons * Cost per gallon = 225 gallon * $3.54/gallon = $796.5

b)

Time taken = Distance travelled/Speed = 2000/50 = 40 hours

Amount that driver will be paid = $34/hr * 40 hr = $1360

c)

Total cots = Gas cost + Driver Cost = $796.5 + $1360 = $2156.5

d)

Miles per gallon at speed of 80 mi/hr = 400/80 miles per gallon = 5 miles per gallon

Number of gallons needed to cover 2000 miles = 2000/5 = 400 gallons

Cost of the gas = Number of gallons * Cost per gallon = 400 gallon * $3.54/gallon = $1416

e)

Time taken = Distance travelled/Speed = 2000/80 = 25hours

Amount that driver will be paid = $34/hr * 25 hr = $850

f)

Total cots = Gas cost + Driver Cost = $1416+ $850= $2266

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