Let's solve this problem step by step.

Part (a) - Find the linear function for the volume of water in the pond.

• The initial amount of water in the pond is $200$ gallons.
• The rate of water being added is $18$ gallons per minute.
• The volume of water in the pond at any time $t$ is a function of the time elapsed, and we can write this function as:

$V(t) = \text{Initial volume} + \text{Rate of change} \times t$

Substitute the values:

$V(t) = 200 + 18t$

Thus, the linear function is:

$V(t) = 200 + 18t$

Part (b) - How long does it take to fill the pond completely?

The total capacity of the pond is $1946$ gallons. To find the time it takes to fill the pond, we set the volume $V(t)$ equal to $1946$ and solve for $t$:

$1946 = 200 + 18t$

Now, solve for $t$:

$1946 - 200 = 18t$

$1746 = 18t$

$t = \frac{1746}{18}$

$t = 97 \, \text{minutes}$

Final Answers:

(a) The linear function for the volume of water is: $V(t) = 200 + 18t$

(b) It takes 97 minutes to completely fill the pond.

Would you like any more details or further explanation?

Here are 5 related questions for practice:

1. What is the rate of water inflow if the hose fills at 25 gal/min?
2. How much water will be in the pond after 50 minutes?
3. If the pond starts with 500 gallons, how does the function change?
4. How long will it take to fill a pond with a capacity of 2500 gallons?
5. What happens if the hose adds water at 18 gal/min but the pond leaks at 5 gal/min?

Tip: In word problems involving rates, pay close attention to initial conditions and how the rate affects the total change over time!