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.

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.