Question

math 1031 chapter r project

this problem is like the chapter r application found in pearson. answer the questions below and then upload your document into canvas using this label: lastname_applicationr

a small pump takes 8 more hours than a large pump to empty. the rate for the large pump to drain the pool can be represented by the rational expression \\(\frac{1}{x}\\) where x is the time that it takes for the large pump to drain the pool.

part 1: a simplified rational expression that describes the combined rate if both pumps are used to drain the pool is: \\(\frac{1}{x} + \frac{1}{x+8}\\). what does the \\(\frac{1}{x+8}\\) represent?

part 2: the combined rate to drain the pool if both pumps are used follows these steps. give a reason for each step. a word bank is provided here:
factor, get common denominators, add, simplify

\

$$\begin{tabular}{|l|l|} \\hline problem step & reason \\\\ \\hline \\(\\frac{1}{x} + \\frac{1}{x+8}\\) & combined rate \\\\ \\hline \\(\\frac{x+8}{x(x+8)} + \\frac{x}{x(x+8)}\\) & \\\\ \\hline \\(\\frac{x+8+x}{x(x+8)}\\) & \\\\ \\hline \\(\\frac{2x+8}{x(x+8)}\\) & \\\\ \\hline \\(\\frac{2(x+4)}{x(x+8)}\\) & \\\\ \\hline \\end{tabular}$$

part 3: what is the combined rate to drain the pool if it takes the larger pump 5 hours to drain the pool? show all work. use correct units in your answer.

Explanation:

Identify given information and goals

We have a work rate problem involving two pumps draining a pool.

• Let \(x\) be the time (in hours) for the large pump to drain the pool.
• The large pump's rate is \(\frac{1}{x}\) pool per hour.
• The small pump takes 8 more hours than the large pump, so its time is \(x + 8\) hours.
• The small pump's rate is \(\frac{1}{x+8}\) pool per hour.
• We need to solve three parts:

1. Explain what \(\frac{1}{x+8}\) represents.
2. Fill in the reasons for each step in the simplification table using the word bank: Factor, Get common denominators, Add, Simplify.
3. Calculate the combined rate if the larger pump takes 5 hours (\(x = 5\)).

Analyze Part 1

The expression \(\frac{1}{x+8}\) represents the rate of the small pump.
Specifically, it is the fraction of the pool that the small pump can drain in one hour.

Analyze Part 2

We match each algebraic step with the correct term from the word bank:

• Step 1: \(\frac{1}{x} + \frac{1}{x+8}\) is given as the "Combined rate".
• Step 2: \(\frac{x+8}{x(x+8)} + \frac{x}{x(x+8)}\)
• Both terms now share a common denominator.
• Corresponding reason: Get common denominators.
• Step 3: \(\frac{x+8+x}{x(x+8)}\)
• The numerators are combined over the single denominator.
• Corresponding reason: Add.
• Step 4: \(\frac{2x+8}{x(x+8)}\)
• The terms in the numerator are combined (\(x + x = 2x\)).
• Corresponding reason: Simplify.
• Step 5: \(\frac{2(x+4)}{x(x+8)}\)
• The numerator \(2x+8\) is written as \(2(x+4)\).
• Corresponding reason: Factor.

Calculate Part 3

We are given that the larger pump takes \(5\) hours to drain the pool, so \(x = 5\).
We substitute \(x = 5\) into the simplified combined rate expression:

$$ \text{Combined Rate} = \frac{2x+8}{x(x+8)} $$

Substituting \(x = 5\):

$$ \text{Combined Rate} = \frac{2(5)+8}{5(5+8)} = \frac{10+8}{5(13)} = \frac{18}{65} $$

The unit of rate is "pools per hour" (or "of the pool per hour").

Answer:

Part 1

The expression \(\frac{1}{x+8}\) represents the rate of the small pump, which is the fraction of the pool that the small pump can drain in one hour.

Part 2

The completed table with reasons from the word bank:

Problem Step	Reason
\(\frac{x+8}{x(x+8)} + \frac{x}{x(x+8)}\)	Get common denominators
\(\frac{x+8+x}{x(x+8)}\)	Add
\(\frac{2x+8}{x(x+8)}\)	Simplify
\(\frac{2(x+4)}{x(x+8)}\)	Factor

Part 3

To find the combined rate when \(x = 5\) hours:

$$ \text{Combined Rate} = \frac{2(5)+8}{5(5+8)} = \frac{18}{65}\text{ pool per hour} $$