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Let's solve the problem in the image step by step.

### (a) Find a linear function \( V(t) \) that models the volume of water in the pond at any time \( t \):

The problem states that:
- The pond initially contains 200 gallons of water.
- Water is being added at a rate of 11 gallons per minute.

This is a classic linear function problem, where the initial value (at \( t = 0 \)) is the starting amount of water, and the slope is the rate at which the water is being added.

The formula for \( V(t) \) would be:
\[
V(t) = \text{{initial volume}} + (\text{{rate of filling}}) \times t
\]
Substituting the given values:
\[
V(t) = 200 + 11t
\]
Thus, the linear function is:
\[
V(t) = 200 + 11t
\]

### (b) If the pond has a capacity of 1146 gallons, how long does it take to completely fill the pond?

To find the time \( t \), we set \( V(t) = 1146 \) and solve for \( t \):
\[
1146 = 200 + 11t
\]
Subtract 200 from both sides:
\[
1146 - 200 = 11t
\]
\[
946 = 11t
\]
Now, divide both sides by 11:
\[
t = \frac{946}{11} = 86
\]

So, it takes **86 minutes** to completely fill the pond.

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Would you like more details or have any questions?

Here are 5 related questions:
1. How would the function change if the rate of filling increased to 15 gallons per minute?
2. What would the function look like if there was a leak draining water at 2 gallons per minute?
3. How much water will be in the pond after 30 minutes?
4. If the initial volume was 0 gallons instead of 200, what would the linear function look like?
5. How would you model the volume if the rate of water filling was not constant?

**Tip**: Linear functions are ideal for problems with constant rates of change. Make sure to identify the slope and initial value when constructing them.

A large koi pond is filled from a garden hose at the rate of 11 gal/min. Initially, the pond contains 200 gal of water. Find a linear function V(t) that models the volume of water in the pond at any time t. If the pond has a capacity of 1146 gal, how long does it take to completely fill the pond?

Learn how to model the volume of water in a koi pond over time using linear functions. This example demonstrates how to set up and solve a linear equation to find out how long it takes to fill a pond that starts with 200 gallons of water and fills at a rate of 11 gallons per minute, with a capacity of 1146 gallons.

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