Linda is pumping water into a tank at a rate of

Question

18
liters per minute. Since she started pumping water,
10
liters has splashed out of the tank. Her goal is given by
≤−18x1062
, where
x
is the number of minutes she pumps water into the tank.
Complete the parts below.
(a)Solve the given inequality and graph the solution on the number line below.

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(b)Choose and complete the statement that best describes the solution.

Tomeetthegoal,thetankwillhaveatleastlitersofwaterinit.Thiswilltakeatmostminutes.
Tomeetthegoal,thetankwillhaveatmostlitersofwaterinit.Thiswilltakeatmostminutes.
Tomeetthegoal,thetankwillhaveatmostlitersofwaterinit.Thiswilltakeatleastminutes.

Answer 1

To solve the given inequality and understand the problem, we start with Linda's water pumping situation.

The inequality provided is:

\[ W \leq -18x + 1062 \]

Where:

• \( W \) is the amount of water in the tank after \( x \) minutes.
• Linda is pumping water at a rate of 18 liters per minute, and 10 liters have splashed out.

The net amount of water in the tank after \( x \) minutes can be calculated as:

\[ W = 18x - 10 \]

Now, we set this expression for \( W \) in relation to the inequality given:

\[ 18x - 10 \leq -18x + 1062 \]

Next, we will solve this inequality.

1. Add \( 18x \) to both sides:

\[ 18x + 18x - 10 \leq 1062 \]

\[ 36x - 10 \leq 1062 \]

2. Add 10 to both sides:

\[ 36x \leq 1072 \]

3. Divide both sides by 36:

\[ x \leq \frac{1072}{36} \]

4. Calculating the fraction gives:

\[ x \leq 29.78 \]

Since \( x \) represents minutes, we can round down to 29 minutes as she can only pump water for whole minutes:

\[ x \leq 29 \]

Now, we will represent the solution on a number line. The solution \( x \leq 29 \) means that she can pump water for any time from 0 to 29 minutes.

Graphing:

On a number line ranging from 0 to 30, we would represent the solution with a closed circle at 29 and shade to the left to indicate all values less than or equal to 29.

 |---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0   1   2   3   4   5   6   7   8   9   10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30 
                                         ●====================================>

(b) Completing the statement:

The solution means that the tank will have at least a certain amount of water in it when Linda pumps water for at most 29 minutes. The correct statement would be:

"To meet the goal, the tank will have at least liters of water in it. This will take at most 29 minutes."

You would need to calculate the total amount of water at \( 29 \) minutes to fill in the answer of how many liters will be in the tank at this point:

\[ W = 18(29) - 10 = 522 - 10 = 512 , \text{liters} \]

So the complete statement could be:

"To meet the goal, the tank will have at least 512 liters of water in it. This will take at most 29 minutes."