Reese is installing an in-ground rectangular pool in her backyard. Her pool will be 30 feet long, 14 feet wide, and have an average depth of 8 feet. She is installing two pipes to bring water to fill the pool; these pipes will also be used to drain the pool at the end of each season. One pipe can fill and drain the pool at a rate that is 1 more than 2 times faster than the other pipe. If both pipes are open and working properly, it will take 3.5 hours to fill the pool. Use this information and what you know about solving rational equations to answer the questions below. 1. When Reese opens the pipes to fill the pool, she finds that only the slower pipe is working. Write a rational equation that you can use to find out how long it will take to fill the pool under these circumstances. Explain how you came up with your equation.

Answer 1

To find the time it will take to fill the pool using only the slower pipe, we can start by calculating the volume of the pool and understanding the rates of each pipe.

1. Calculate the volume of the pool: The formula for the volume \(V\) of a rectangular prism (like a pool) is: \[ V = \text{length} \times \text{width} \times \text{average depth} \] Given:

   • Length = 30 feet
   • Width = 14 feet
   • Average depth = 8 feet

   Plugging in the numbers: \[ V = 30 , \text{ft} \times 14 , \text{ft} \times 8 , \text{ft} = 3360 , \text{ft}^3 \]

2. Define the rates of the pipes: Let the rate of the slower pipe be \(r\) (in cubic feet per hour). According to the problem, the faster pipe fills at a rate that is \(1 + 2r\).

3. Use the information about both pipes working together: When both pipes are working together, they fill the pool in 3.5 hours. Therefore, the combined rate of the two pipes can be formulated as follows: \[ \text{Combined rate} = r + (1 + 2r) = 3r + 1 , \text{ft}^3/\text{hr} \] Thus, the equation relating the combined rate and the volume of the pool is: \[ (3r + 1) \cdot 3.5 = 3360 \] This can be simplified to: \[ 3r + 1 = \frac{3360}{3.5} \] Evaluating the right side gives: \[ 3r + 1 = 960 \] Therefore, we can solve for \(r\): \[ 3r = 960 - 1 \implies 3r = 959 \implies r = \frac{959}{3} \approx 319.67 , \text{ft}^3/\text{hr} \] This gives us the rate of the slower pipe.

4. Equation to fill the pool using only the slower pipe: Now if only the slower pipe is used, it will fill the pool at a rate of \(r\). The time \(t\) it will take to fill the pool can be formulated as: \[ r \cdot t = 3360 \] This can be rewritten to find \(t\): \[ t = \frac{3360}{r} \] Where \(r\) is the rate we obtained earlier.

5. Final Rational Equation: Therefore, the final rational equation, using the slower pipe rate \(r\), can be written as: \[ t = \frac{3360}{r} \] This equation provides a way to find out how long it will take to fill the pool using only the slower pipe by substituting the value of \(r\) we calculated.

Answer 2

Solve the equation that you wrote in question 1. You can round your final answer(s) to the nearest hundredth. Show all your work.

Answer 3

To solve the equation from question 1, we know that we defined the time \(t\) it will take to fill the pool using only the slower pipe as:

\[ t = \frac{3360}{r} \]

From the earlier part, we found \(r\) (the rate of the slower pipe) using the combined rate of both pipes. The combined rate equation resulted in the following equation:

\[ 3r + 1 = 960 \]

Now, solving for \(r\):

1. Subtract 1 from both sides: \[ 3r = 960 - 1 \] \[ 3r = 959 \]

2. Divide by 3: \[ r = \frac{959}{3} \] \[ r \approx 319.67 \text{ ft}^3/\text{hr} \]

Calculating the time \(t\):

Now that we have \(r\), we can substitute it back into the equation for \(t\):

\[ t = \frac{3360}{r} \] Substituting the value of \(r\):

\[ t = \frac{3360}{319.67} \]

Now let's calculate \(t\):

\[ t \approx 10.50 \text{ hours} \]

Conclusion:

After rounding to the nearest hundredth, the time it will take to fill the pool using only the slower pipe is approximately:

\[ \boxed{10.50} \text{ hours} \]

Answer 4

. Interpret your answer(s) to question 2 in context of the problem. What do the number(s) represent?

Answer 5

In the context of the problem, the answer we calculated—approximately 10.50 hours—represents the time it would take to fill Reese's rectangular pool using only the slower filling pipe.

Interpretation of the Result:

1. Filling the Pool: The time of 10.50 hours indicates that if Reese has no access to the faster pipe, and can only use the slower pipe that we defined with the rate of approximately 319.67 cubic feet per hour, it will take her about 10 hours and 30 minutes to completely fill the pool to its capacity of 3360 cubic feet.

2. Practical Considerations: This information is essential for Reese as it impacts her planning. For instance, if she intends to have the pool filled by a certain time for an event or activity, she would need to start filling the pool early enough based on this estimate.

3. Impact of the Pipe Rate: The slower filling rate directly impacts the total time required. If the faster pipe were functional, the combined rate of both pipes working together (which allowed filling in just 3.5 hours) makes it clear how significant the improvement can be when both pipes are available.

Overall, this answer helps Reese understand the implications of using only one pipe for filling her pool and assists her in making informed decisions about her pool installation and maintenance schedule.