Rational Expression word problem

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Discussion Overview

The discussion revolves around formulating a rational expression for profit per hour based on Nancy's lawn care business, specifically focusing on the relationship between profit, area of the lawn, and time taken for maintenance. Participants explore the mathematical representation of these relationships and evaluate the expression for a specific area.

Discussion Character

  • Mathematical reasoning, Homework-related, Technical explanation

Main Points Raised

  • One participant introduces the profit function as $5A$ and the time function as $0.05A + 0.25$, prompting the need for a profit per hour function.
  • Another participant explains that "per" indicates a division of total profit by the number of hours, suggesting the construction of a profit per hour function $P_h(A) = \frac{5A}{0.05A + 0.25}$.
  • There is a discussion about the importance of using bracketing symbols to clarify the denominator in the expression, with emphasis on correct notation.
  • Participants express uncertainty about the formulation and evaluation of $P_h(90)$, with one participant attempting to clarify the correct expression and its evaluation.
  • Several participants reiterate the need for proper bracketing to avoid misinterpretation of the expression.

Areas of Agreement / Disagreement

Participants generally agree on the structure of the profit per hour function but express uncertainty regarding the correct evaluation of the expression for a specific area. There is no consensus on the final evaluation steps.

Contextual Notes

Participants highlight the importance of notation and bracketing in mathematical expressions, indicating potential confusion in interpreting the expression without clear symbols. There are unresolved aspects regarding the evaluation process for $P_h(90)$.

Pikachu1
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Nancy's lawn care specializes in residential lawn cutting and fertilizing. Nancy the owner has tracked her income and expenses. She has determined that her profit can be represented by the expression $5A, where A is the area of the lawn in square metres. The time in hours that it takes her to maintain a lawn is approximately 0.05A + 0.25.

Write a rational expression for nancy profit per hour.
Evaluate the expression for a yard with an area of 90m^2Obviously there is some sort of formula for this that i don't understand.
 
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When you see the word "per" (which means "for each"), as in miles per hour (or profit per hour), this means you want to take some total and divide by the number of hours. You are given both the total profit, and the number of hours, both as a function of $A$. Can you now construct a profit per hour function $P_h$ for Nancy?

$$P_h(A)=\frac{\text{Total profit}}{\text{hours to maintain a lawn}}=?$$
 
MarkFL said:
When you see the word "per" (which means "for each"), as in miles per hour (or profit per hour), this means you want to take some total and divide by the number of hours. You are given both the total profit, and the number of hours, both as a function of $A$. Can you now construct a profit per hour function $P_h$ for Nancy?

$$P_h(A)=\frac{\text{Total profit}}{\text{hours to maintain a lawn}}=?$$
So it would be,

Ph(A) =$5A/ 0.05A + 0.25 ?

I feel like i am still missing something.
 
Pikachu said:
So it would be,

Ph(A) =$5A/ 0.05A + 0.25 ?

I feel like i am still missing something.

I would just recognize that the units of the function are dollars per hour, and if using plain text, use bracketing symbols to indicate clearly what the denominator is as follows:

P_h(A) = 5A/(0.05A + 0.25)

So, for part b), you need to evaluate P_h(90). :D
 
MarkFL said:
I would just recognize that the units of the function are dollars per hour, and if using plain text, use bracketing symbols to indicate clearly what the denominator is as follows:

P_h(A) = 5A/(0.05A + 0.25)

So, for part b), you need to evaluate P_h(90). :D

So, it would be? I have no idea why this question stomped me like that.
P_h(90) = 5(90)/0.05(90) +0.25 ?
 
Pikachu said:
So, it would be? I have no idea why this question stomped me like that.
P_h(90) = 5(90)/0.05(90) +0.25 ?

Well, you really need bracketing symbols...

P_h(90) = 5(90)/(0.05(90) + 0.25)

The way you write it means:

$$P_h(90)=\frac{5(90)}{0.05(90)}+0.25$$

But what you want is:

$$P_h(90)=\frac{5(90)}{0.05(90)+0.25}$$

And the parentheses makes the meaning clear. :D
 
MarkFL said:
Well, you really need bracketing symbols...

P_h(90) = 5(90)/(0.05(90) + 0.25)

The way you write it means:

$$P_h(90)=\frac{5(90)}{0.05(90)}+0.25$$

But what you want is:

$$P_h(90)=\frac{5(90)}{0.05(90)+0.25}$$

And the parentheses makes the meaning clear. :D

Thank you! I understand now.
You rock
 

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