Let's go through each part of the problem step by step:

(a) Find the average rate at which the tank is filled.

The average rate of filling is the change in the amount of water in the tank divided by the change in time.

Given:

• After 3 minutes: 60 gallons
• After 7 minutes: 100 gallons

The change in water = $100 \, \text{gallons} - 60 \, \text{gallons} = 40 \, \text{gallons}$
The change in time = $7 \, \text{minutes} - 3 \, \text{minutes} = 4 \, \text{minutes}$

So, the average rate is: $\text{Average rate} = \frac{40 \, \text{gallons}}{4 \, \text{minutes}} = 10 \, \text{gallons per minute}$

(b) Write an equation that models the amount of water $W(t)$ in the tank as a function of time $t$ (in minutes).

The equation of a line can be written in the form: $W(t) = mt + b$ where $m$ is the slope (rate at which the tank is filled) and $b$ is the initial amount of water in the tank (when $t = 0$).

From part (a), the slope $m = 10 \, \text{gallons per minute}$.

We can use one of the points, say $(t, W(t)) = (3, 60)$, to find $b$.

$60 = 10(3) + b$ $60 = 30 + b$ $b = 30 \, \text{gallons}$

So, the equation is: $W(t) = 10t + 30$

(c) How much water was in the tank initially?

The initial amount of water in the tank is the value of $W(t)$ when $t = 0$.

Using the equation $W(t) = 10t + 30$:

$W(0) = 10(0) + 30 = 30 \, \text{gallons}$

So, the tank initially had 30 gallons of water.

(d) How long did it take to fill the tank to 90 gallons?

We need to find the time $t$ when $W(t) = 90$ gallons.

Using the equation $W(t) = 10t + 30$:

$90 = 10t + 30$ $90 - 30 = 10t$ $60 = 10t$ $t = \frac{60}{10} = 6 \, \text{minutes}$

It took 6 minutes to fill the tank to 90 gallons.

Summary of Answers:

(a) The average rate is 10 gallons per minute.
(b) The equation modeling the amount of water in the tank is $W(t) = 10t + 30$.
(c) Initially, the tank had 30 gallons of water.
(d) It took 6 minutes to fill the tank to 90 gallons.

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. If the tank continues to fill at the same rate, how much water will be in the tank after 15 minutes?
2. How long will it take to fill the tank to 150 gallons?
3. What would be the rate if it took 5 minutes to fill the tank from 30 to 80 gallons?
4. If the initial amount of water was 50 gallons, what would be the new equation for $W(t)$?
5. How much water will be in the tank after 12 minutes?

Tip: Always check that the units are consistent throughout the problem when calculating rates or solving equations.