An employee of a delivery company earns 10 per hour driving...

An employee of a delivery company earns $\$ 10$ per hour driving a delivery van in an area where gasoline costs $\$ 2.80$ per gallon. When the van is driven at a constant speed $s 2.80$ per hour, with $40 \leq s \leq 65$, the van gets $700 / s$ miles per gallon. (a) Find the cost $C$ as a function of $s$ for a 100-mile trip on an interstate highway. (b) Use a graphing utility to graph the function found in part (a) and determine the most economical speed.

An employee of a delivery company earns $\$ 10$ per hour driving a delivery van in an area where gasoline costs $\$ 2.80$ per gallon. When the van is driven at a constant speed $s 2.80$ per hour, with $40 \leq s \leq 65$, the van gets $700 / s$ miles per gallon.
(a) Find the cost $C$ as a function of $s$ for a 100-mile trip on an interstate highway.
(b) Use a graphing utility to graph the function found in part (a) and determine the most economical speed.

Key Concepts

Cost Function Modeling

This concept involves formulating a mathematical function that represents the total cost associated with an activity—in this case, a trip. It combines various cost elements, such as fuel cost and labor cost, into a single equation dependent on parameters like speed and distance. The cost function enables one to analyze how changes in input variables affect the overall expenditure.

Rate Problems and Unit Analysis

Rate problems deal with quantities that change with respect to time or other variables, such as miles per hour or dollars per gallon. Unit analysis is used to ensure that the calculations correctly combine these diverse rates (for instance, combining cost per hour with cost per mile) to create a coherent function. Understanding these rates is crucial for constructing accurate models in transportation and economics.

Graphical Analysis and Optimization

Graphical analysis involves using tools or methods to visually represent functions, which aids in identifying features like maximums, minimums, or inflection points. Optimization is the process of finding the best solution under given constraints—in this case, finding the speed that minimizes the total cost. Together, these concepts often lead to identifying optimal operational parameters in real-world applications.

Constraint and Domain Considerations

When modeling real-life situations, it is important to account for practical constraints and limitations of the variables involved. This includes defining the domain of the function—such as the range of speeds for safe or legal operation. Incorporating these constraints ensures that the solution is both mathematically sound and practically applicable.

Fuel Efficiency Modeling

Fuel efficiency modeling is about expressing how effectively a vehicle uses fuel, often as a function of speed or other operating conditions. In optimization problems, fuel consumption is typically modeled as either directly proportional or inversely proportional to variables like speed. This allows for the careful analysis of fuel use relative to performance, which is essential in optimizing operational costs.

Transcript

00:01 This is kind of a fun little problem, although it's kind of, i wasn't exactly sure what they wanted.

00:08 So they tell us an employee of a delivery company earns $10 an hour.

00:12 So i said, okay, that's his rate, his wage rate.

00:17 A driving delivery van in an area where gasoline costs $200 .80 per gallon.

00:23 So this is the cost of gas per gallon.

00:27 They say that in at a constant speed when the drive van is driven in a constant speed s within s between between 40 and 65 miles an hour it gets a it gets a gas mileage of 700 over s miles per gallon so this is the miles per gallon so this is the gas mileage it gets and so then they ask us find the cost as a function of s for a mile trip on an interstate highway so i think what they mean is is the net cost so um at first i did this problem i was thinking it was basically looking at the driver so you know the driver has a cost um and that's just the amount of gas so that's why i called this c but then i realize they're probably they're talking about the delivery company, i think, the cost of the delivery company.

01:25 So that's what i then call down here, the net cost.

01:29 So the cost, this is the cost of the gas.

01:33 Maybe i should have called that capital g or something.

01:36 And so he goes 100 miles.

01:39 So the cost of the gas is $2 .80 per gallon, divided by the gas mileage, 700 divided by s miles per gallon, times 100 miles, and that gives you a cost of 40 cents times s.

02:01 So what it says is that the faster he goes, the greater the cost is going to be linearly.

02:11 And that's expected because the gas mileage is going to go.

02:14 Go down.

02:17 But the faster he goes, the less time it takes.

02:21 So if the company is paying him this rate, you can see how much money they're going to have to pay the delivery driver.

02:28 So the time that the deliver driver is working is 100, well, we get the distance divided by the speed.

02:36 So that's 100 divided by the speed hours.

02:40 And so his rate is $10 an hour.

02:42 So the total money that he makes is $10 per hour times 100 divided by s hours or 1 ,000 divided by s dollars.

02:53 And so the total cost, and i think, did i miss a minus sign in there? no, i should have missed a plus sign here...

Please give Ace some feedback