A faucet used to add water to a large bottle that already continued some water. After it has been filling for 3 seconds, the gauge on the bottle indicates that it contains 20 ounces of water. After it has been filling for 11 seconds, the gauge indicates the bottle contains 60 ounces of water. Let y be  the amount of water in the bottle x seconds after the faucet was turned on. Give a linear equation (in slope intercept form) that models the amount of water in the bottle in terms of x.

We are given two points of dataWe are given two points of  

- After 3 seconds, the bottle contains 20 ounces of water.
- After 11 seconds, the bottle contains 60 ounces of water.

We need to find the linear equation in slope-intercept form \( y = mx + b \), where:
- \( y \) is the amount of water in the bottle after \( x \) seconds.
- \( m \) is the slope (rate of water flow).
- \( b \) is the y-intercept (initial amount of water in the bottle when \( x = 0 \)).

### Step 1: Calculate the slope \( m \)
The slope \( m \) represents the rate of water flow, or the change in the amount of water per second. We can find \( m \) using the formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \):

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \( (3, 20) \) and \( (11, 60) \):

\[
m = \frac{60 - 20}{11 - 3} = \frac{40}{8} = 5
\]

So, the slope \( m = 5 \), which means the water is flowing at a rate of 5 ounces per second.

### Step 2: Use the slope to find the y-intercept \( b \)
We now use one of the given points and the slope to find \( b \). Let's use the point \( (3, 20) \) and substitute into the equation \( y = mx + b \):

\[
20 = 5(3) + b
\]

\[
20 = 15 + b
\]

\[
b = 5
\]

So, the y-intercept \( b = 5 \). This means there were initially 5 ounces of water in the bottle when the faucet was turned on.

### Step 3: Write the equation
Now that we have the slope \( m = 5 \) and the y-intercept \( b = 5 \), the equation of the line is:

\[
y = 5x + 5
\]

This is the linear equation that models the amount of water in the bottle after \( x \) seconds.

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Would you like further details or have any questions?

### 5 related questions:
1. How would the equation change if the initial amount of water was different?
2. What is the amount of water in the bottle after 15 seconds?
3. How long would it take for the bottle to reach 100 ounces?
4. What does the slope represent in the context of this problem?
5. How would the equation be affected if the faucet's flow rate increased?

### Tip:
When working with linear models, the slope represents the rate of change, and the y-intercept gives the starting value of the system being modeled.

A faucet used to add water to a large bottle that already contained some water. After it has been filling for 3 seconds, the gauge on the bottle indicates that it contains 20 ounces of water. After it has been filling for 11 seconds, the gauge indicates the bottle contains 60 ounces of water. Let y be the amount of water in the bottle x seconds after the faucet was turned on. Give a linear equation (in slope intercept form) that models the amount of water in the bottle in terms of x.

Learn how to model the amount of water in a bottle over time using a linear equation in slope-intercept form. This problem explores how to calculate the slope and y-intercept, based on data showing the bottle contains 20 ounces after 3 seconds and 60 ounces after 11 seconds. Ideal for students in Grades 8-10.

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