# Find linear equation based on given costs

• ducmod
In summary, the conversation discusses solving a problem involving a taxi company that charges $2.50 for the first 1/5 of a mile and$0.45 for each additional 1/5 of a mile. The desired output is a linear function that gives the correct fee for distances that are even multiples of 1/5 of a mile. The solution involves finding a point (1/5, 2.50) and a slope of 2.25, resulting in a function of y = 2.25x + 2.05. However, this solution fails for trips between 0 and 1/5 of a mile, leading to the suggestion of defining the solution piecewise. The conversation also suggests
ducmod

## Homework Statement

Hello!

Please, take a look at the screeshot with a problem description.

## Homework Equations

I am trying to solve this, but I seem to have a wrong understanding of the problem.

## The Attempt at a Solution

It is said that a company charges 2.50 for the first 1/5 of a mile; therefore, depending on how many miles a passenger has ridden, it should be 2.50 x m for every mile (every first fifth of every mile ), where m = miles.

Than, it is said that every additional 1/5 costs 0.45. This means that 0.45 x m x n, where m = miles and n = number of fifths in each mile; or it can be 0.45 x n, where n = all fifth of all miles during the ride (the last mile might not be complete, thus it will not contain 4/5 as all full miles would).

The answer in the textbook is
F(m) = 2:25m+2:05 The slope 2:25 means it costs an additional $2:25 for each mile beyond the rst 0.2 miles. F(0) = 2:05, so according to the model, it would cost$2:05 for a trip of 0
miles.

How 2.25 has been computed? Why is there 2.05 (which I assume is 2.50 - 0.45)?

Thank you very much!

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ducmod said:

## Homework Equations

I am trying to solve this, but I seem to have a wrong understanding of the problem.

## The Attempt at a Solution

It is said that a company charges 2.50 for the first 1/5 of a mile; therefore, depending on how many miles a passenger has ridden, it should be 2.50 x m for every mile (every first fifth of every mile ), where m = miles.
The company charges 2.50 for the first 1/5 of the first mile in the trip. If the passenger is driven 1/5 of a mile, the total fee for the trip will be $2.50. That$2.50 fee is incurred only once per trip regardless of trip length.

Than, it is said that every additional 1/5 costs 0.45. This means that 0.45 x m x n, where m = miles and n = number of fifths in each mile; or it can be 0.45 x n, where n = all fifth of all miles during the ride (the last mile might not be complete, thus it will not contain 4/5 as all full miles would).
All full miles other than the first involve fees for five fifths, not four.

The problem is not expecting you to delve into the details of what happens between each fifth of a mile -- it is not asking you to come up with a step function. Instead it asks for a linear function. I would suggest that you come up with a linear function which gives the correct fee for distances that are even multiples of 1/5 of a mile.

You appear to be misunderstanding the term "first fifth of a mile" apparently interpreting that to mean that, for every mile, you are charged $2.50 for the first 1/5 mile, then$0.45 for each of the next 1/5 of a mile in that mile for a total of $2.50+ 4(0.45)=$4.30 per mile. That is wrong. The "first fifth of a mile" means the first 1/5 mile of the journey. If you ride in the taxi for 4 miles, say, that would be a total of 4(6)= 24 "1/5 miles". You would pay $2.50 for the first 1/5 mile then$0.45 for each of the other 23 "1/5 miles" for a total of $2.50+ 23(0.45)=$11.75.

Thank you, Friends, for your help!
I got it. Though I still think that the problem is ambiguously defined.

I solved it this way:

0.45 x 5 = 2.25

Therefore, we have a point (1/5 ; 2.50 ) and a slope of 2.25. Solving for y, we get y = 2.25m - 2.25 x (1/5) + 2.50 = 2.25m + 2.05

But it is still not clear to me, why 2.05 is a constant, if 2.50 is paid for the first 1/5 of a mile, and 2.50 is constant. In terms of usual meaning, 2.05 would mean a fixed cost that has to be paid even if the ride took zero miles.

ducmod said:
But it is still not clear to me, why 2.05 is a constant,
What? Any specific number cannot be anything but a constant! And where did you get "2.05"? I'm not clear now whether your difficulty is with mathematics or the English language.

if 2.50 is paid for the first 1/5 of a mile, and 2.50 is constant. In terms of usual meaning, 2.05 would mean a fixed cost that has to be paid even if the ride took zero miles.

ducmod said:
But it is still not clear to me, why 2.05 is a constant, if 2.50 is paid for the first 1/5 of a mile, and 2.50 is constant. In terms of usual meaning, 2.05 would mean a fixed cost that has to be paid even if the ride took zero miles.
Your solution works for journeys longer than 1/5 of a mile. But fails for trips between 0 and 1/5 of a mile. Try plotting the cost...it is not linear. It is piecewise linear.
Maybe you should consider defining the solution piecewise:
For x in miles:
##f(x) = \left\{ \begin{array}{l l} ? & 0 \leq x < 1/5 \\
? &1/5 \leq x \end{array} \right. ##

0 miles: Not applicable. No passengers will embark on or pay for such trips. Price is, accordingly, irrelevant.
1/5 mile: Pay $2.50 for first 1/5 mile for a total of$2.50.
2/5 miles: Pay $2.50 for first 1/5 mile and$0.45 for second 1/5 mile for a total of $2.95 3/5 miles: Pay$2.50 for first 1/5 mile and $0.90 for the succeeding 2/5 mile for a total of$3.40

Extrapolate to a linear function that is correct at each even multiple of 1/5 mile.

## 1. How do I find the linear equation given a set of costs?

To find the linear equation based on given costs, you will need to have at least two data points (x,y pairs) where x represents the quantity and y represents the cost. Once you have these data points, use the slope formula (m = (y2 - y1)/(x2 - x1)) to calculate the slope and the y-intercept formula (b = y - mx) to calculate the y-intercept. The linear equation will be in the form of y = mx + b, where m is the slope and b is the y-intercept.

## 2. Can I use any two data points to find the linear equation?

No, you will need to make sure that the two data points you use are accurate and representative of the relationship between quantity and cost. If the data points are not accurate, the resulting linear equation will not accurately represent the relationship and may lead to incorrect conclusions.

## 3. What is the significance of finding the linear equation based on given costs?

Finding the linear equation allows you to understand the relationship between quantity and cost. It can help you make predictions and decisions based on the data, such as determining the most cost-effective quantity to produce, or forecasting future costs based on projected quantities.

## 4. Can I use the linear equation to find the cost for a specific quantity?

Yes, once you have the linear equation, you can plug in a specific quantity value for x and solve for the corresponding cost value for y. This allows you to make accurate cost predictions for any quantity within the range of the data points used to create the linear equation.

## 5. Is there a specific method for finding the linear equation based on given costs?

Yes, as mentioned earlier, you will need to use the slope and y-intercept formulas to calculate the values for m and b, which can then be used to create the linear equation. Alternatively, you can use a graphing calculator or a spreadsheet program to plot the data points and automatically generate the linear equation.

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