An employee of a delivery company earns $\$$ 25.00 per hour driving a delivery van in an area where gasoline costs $\$$ 2.90 per gallon. When the van is driven at a constant speed s (in miles per hour, with $45 \leq s \leq 60$ ), the van gets $\frac { 290 } { s }$ 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

Quizplus · Accepted Answer

To find the most economical speed, we need to minimize the total cost of the trip, which includes both the cost of gasoline and the cost of the employee's time. The total cost (C) for a 100-mile trip can be expressed as the sum of the gasoline cost and the wage cost. The gasoline cost is $\$2.90$ per gallon times the number of gallons used, which is $100$ miles divided by the fuel efficiency $\frac{290}{s}$ miles per gallon. The wage cost is $\$25.00$ per hour times the number of hours, which is $100$ miles divided by $s$ miles per hour. So, the total cost $C$ as a function of speed $s$ is:$$C(s) = \left(\frac{\$2.90 \times 100}{\frac{290}{s}}\right) + \left(\frac{\$25.00 \times 100}{s}\right)$$Simplifying gives:$$C(s) = \left(\$2.90 \times \frac{s}{2.9}\right) + \left(\frac{\$2500}{s}\right)$$$$C(s) = \$s + \frac{\$2500}{s}$$To find the minimum cost, we would take the derivative of $C(s)$ with respect to $s$, set it to zero, and solve for $s$. However, without going through the calculus, we can reason that the function has a decreasing cost up to a certain point (speed) and then starts increasing, indicating a minimum cost at some speed within the given range.Given the choices, we can evaluate or approximate the cost for each speed or recognize that the formula suggests a balance between speed and efficiency. The correct answer, based on minimizing the total cost considering both fuel efficiency and wage cost, is:D) The most economical speed is 50.0 mph.

An Employee of a Delivery Company Earns $\$$ 2500 Per Hour Driving a Delivery Van in an Area