Question

# For speeds between 40 mph and 65 mph, a diesel truck gets 460x460x miles per gallon...

For speeds between 40 mph and 65 mph, a diesel truck gets 460x460x miles per gallon when driven at a constant speed of xx miles per hour. Diesel gasoline costs 4 dollars per gallon. The driver is paid 22 dollars per hour.

A) What is the total cost to have the truck driven 100 miles at a constant speed of 60 miles per hour?
dollars.

B) What is the best constant speed to drive the truck to minimize the total cost to drive it 100 miles?
miles per hour.

C) How much money is saved, per 100 miles, by driving at the the constant speed you found in problem B) instead of 60 miles per hour?
dollars.

Solution:

a)

At speed of 60 miles per hour, the diesel truck will give average of 460/60 = 46/6 miles per gallon Time taken for journey =100/60 hours = 5/3 hours

Total cost = Cost of diesel + Cost of driver

4 ( 600/46 ) + 22 * 5 / 3

\$ 88.84

b)

Let the constant speed be x

Time taken = 100/x

Total Cost  = Cost of diesel + Cost of driver

4 (100/(460/x)) + 22 * 200/x

400 x / 460 + 2200/x

Taking the derivative

we get

400/460 - 2200 / x2 = 0

x = = 50.29

Hence the be. constant speed is 50. 29 mph

C)

Cost with speed of 50.29 will be

400 * 50.29 / 460 + 2200 / 50.29 = 87.47

Money saved = 88.84 - 87.47

= \$1.37

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