An employee of a delivery company earns $25.00 per hour driving a delivery van in an area where gasoline costs $2.60 per gallon.When the van is driven at a constant speed s (in miles per hour,with ),the van gets miles per gallon.Determine the most economical speed s for a 100-mile trip on an interstate highway.
A) The most economical speed is 52.0 mph.
B) The most economical speed is 48.0 mph.
C) The most economical speed is 25.0 mph.
D) The most economical speed is 50.0 mph.
E) The most economical speed is 53.0 mph.

Question

​An employee of a delivery company earns $25.00 per hour driving a delivery van in an area where gasoline costs $2.60 per gallon.When the van is driven at a constant speed s (in miles per hour,with  ),the van gets  miles per gallon.Determine the most economical speed s for a 100-mile trip on an interstate highway. ​
A) ​The most economical speed is 52.0 mph. 
B) ​The most economical speed is 48.0 mph. 
C) ​The most economical speed is 25.0 mph. 
D) ​The most economical speed is 50.0 mph. 
E) ​The most economical speed is 53.0 mph.

Quizplus · Accepted Answer

To find the most economical speed, we need to minimize the total cost of the trip (fuel cost + driver's wages). Let's call the speed s the van drives at in miles per hour.

First, we need to find how many gallons of gasoline the van will use for the 100-mile trip. Since the van gets 11ea8970_6a42_da45_861b_313b22a02994_TB1301_00 miles per gallon, it will use 100/11ea8970_6a42_da45_861b_313b22a02994_TB1301_00 = 9.0909... gallons of gasoline.

Next, we need to find how much the gasoline will cost. Since gasoline costs $2.60 per gallon, the cost will be 2.60 x 9.0909... = $23.64 (rounded to the nearest penny).

Finally, we need to find how much the driver's wages will cost. Since the trip takes 100/s hours, the cost will be 25 x 100/s = 2500/s.

Therefore, the total cost C of the trip will be:

C = 23.64 + 2500/s

To find the value of s that minimizes C, we need to take the derivative of C with respect to s and set it equal to 0:

dC/ds = -2500/s^2 = 0

Solving for s, we get s = sqrt(2500/ea8970_6a42_da45_861b_a929b55a2ff2_TB1301_00) = 50 mph (rounded to the nearest mile per hour).

Therefore, the most economical speed is 50 mph.

​An Employee of a Delivery Company Earns $25

An Employee of a Delivery Company Earns $25