Question

time (minutes) | water (gallons)
6 | 112
9 | 149
12 | 184
15 | 220
answer the following questions.
(a) choose the statement that best describes how the time and the amount of water in the pool are related. then give the value requested.

• as time increases, the amount of water in the pool decreases.

at what rate is the amount of water in the pool decreasing? (square) gallons per minute

• as time increases, the amount of water in the pool increases.

at what rate is the amount of water in the pool increasing? (square) gallons per minute
(b) how much water was already in the pool when nicole started adding water? (square) gallons

Explanation:

Part (a)

Step 1: Determine the relationship (increasing or decreasing)

We have time (minutes) and water (gallons) data: (6, 112), (9, 148), (12, 184), (15, 220). As time increases from 6 to 9 to 12 to 15 minutes, the water gallons increase from 112 to 148 to 184 to 220. So we choose "As time increases, the amount of water in the pool increases."

Step 2: Calculate the rate of increase

To find the rate (slope), we use the formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\): \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take the first two points \((6, 112)\) and \((9, 148)\). Then \(m=\frac{148 - 112}{9 - 6}=\frac{36}{3} = 12\) gallons per minute. We can check with other points: \((9, 148)\) and \((12, 184)\): \(\frac{184 - 148}{12 - 9}=\frac{36}{3}=12\), and \((12, 184)\) and \((15, 220)\): \(\frac{220 - 184}{15 - 12}=\frac{36}{3}=12\). So the rate of increase is 12 gallons per minute.

Part (b)

Step 1: Find the equation of the line

We know the slope \(m = 12\) (from part a) and we can use the point - slope form \(y - y_1=m(x - x_1)\). Let's use the point \((6, 112)\), where \(x_1 = 6\), \(y_1=112\) and \(m = 12\).

The point - slope form is \(y-112 = 12(x - 6)\)

Step 2: Find the initial amount (when \(x = 0\))

To find the amount of water when Nicole started (at \(x = 0\) minutes), we substitute \(x = 0\) into the equation:

\(y-112=12(0 - 6)\)

\(y-112=12\times(- 6)\)

\(y-112=-72\)

\(y=112 - 72\)

\(y = 40\)

Final Answers

(a) The correct statement is "As time increases, the amount of water in the pool increases." The rate of increase is \(\boldsymbol{12}\) gallons per minute.

(b) The amount of water already in the pool when Nicole started adding water is \(\boldsymbol{40}\) gallons.

Answer:

Part (a)

Step 1: Determine the relationship (increasing or decreasing)

We have time (minutes) and water (gallons) data: (6, 112), (9, 148), (12, 184), (15, 220). As time increases from 6 to 9 to 12 to 15 minutes, the water gallons increase from 112 to 148 to 184 to 220. So we choose "As time increases, the amount of water in the pool increases."

Step 2: Calculate the rate of increase

To find the rate (slope), we use the formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\): \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take the first two points \((6, 112)\) and \((9, 148)\). Then \(m=\frac{148 - 112}{9 - 6}=\frac{36}{3} = 12\) gallons per minute. We can check with other points: \((9, 148)\) and \((12, 184)\): \(\frac{184 - 148}{12 - 9}=\frac{36}{3}=12\), and \((12, 184)\) and \((15, 220)\): \(\frac{220 - 184}{15 - 12}=\frac{36}{3}=12\). So the rate of increase is 12 gallons per minute.

Part (b)

Step 1: Find the equation of the line

We know the slope \(m = 12\) (from part a) and we can use the point - slope form \(y - y_1=m(x - x_1)\). Let's use the point \((6, 112)\), where \(x_1 = 6\), \(y_1=112\) and \(m = 12\).

The point - slope form is \(y-112 = 12(x - 6)\)

Step 2: Find the initial amount (when \(x = 0\))

To find the amount of water when Nicole started (at \(x = 0\) minutes), we substitute \(x = 0\) into the equation:

\(y-112=12(0 - 6)\)

\(y-112=12\times(- 6)\)

\(y-112=-72\)

\(y=112 - 72\)

\(y = 40\)

Final Answers

(a) The correct statement is "As time increases, the amount of water in the pool increases." The rate of increase is \(\boldsymbol{12}\) gallons per minute.

(b) The amount of water already in the pool when Nicole started adding water is \(\boldsymbol{40}\) gallons.