21. A tank initially has 420 gallons of water. Water is getting pumped out of this tank at the
rate 3 gallons/minute.
(a) Write a function that relates the volume of the water to time.
(b) How long will it take for the tank to be empty?
22. A table of values of a linear function is given. Find a formula for the function in the form
f(x) = ax+b.
x f(x)
2   5
4  8
23. Toms weekly pay is a linear function of the number of hours he worked that week. When
he worked 40 hours on a week his pay was $376. When he worked 50 hours another week, the
pay was $390.
(a) Find Tom’s pay as a function of the number of hours he worked.
(b) Does the pay increase as hours increase? If yes, at what rate?

Let's break these problems down step by step:

---

### 21. Tank problem

#### (a) Write a function that relates the volume of the water to time.

The tank initially has 420 gallons of water, and water is being pumped out at 3 gallons per minute. Let $ V(t) $ represent the volume of water in the tank at time $ t $ (in minutes).

The rate of change of water in the tank is constant, so this is a linear function. The volume of the tank decreases by 3 gallons for every minute that passes, starting at 420 gallons. Therefore, the function can be written as:

\[
V(t) = 420 - 3t
\]

Where:
- $ V(t) $ is the volume of water at time $ t $,
- $ 420 $ is the initial amount of water,
- $ 3 $ is the rate at which water is pumped out (in gallons per minute),
- $ t $ is time in minutes.

#### (b) How long will it take for the tank to be empty?

The tank will be empty when $ V(t) = 0 $. Set the function equal to 0 and solve for $ t $:

\[
420 - 3t = 0
\]
\[
3t = 420
\]
\[
t = \frac{420}{3} = 140
\]

It will take 140 minutes for the tank to be empty.

---

### 22. Linear function from a table

The table of values is:

\[
\begin{array}{c|c}
x & f(x) \\
\hline
2 & 5 \\
4 & 8 \\
\end{array}
\]

The function $ f(x) = ax + b $ is linear, so we need to find the slope $ a $ and the intercept $ b $.

#### Step 1: Find the slope $ a $

The slope $ a $ is the rate of change of $ f(x) $, which is given by:

\[
a = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{8 - 5}{4 - 2} = \frac{3}{2}
\]

#### Step 2: Find the intercept $ b $

Using the point $ (x_1, f(x_1)) = (2, 5) $ and the equation $ f(x) = ax + b $, substitute the values and solve for $ b $:

\[
5 = \frac{3}{2}(2) + b
\]
\[
5 = 3 + b
\]
\[
b = 5 - 3 = 2
\]

Thus, the formula for the function is:

\[
f(x) = \frac{3}{2}x + 2
\]

---

### 23. Tom's weekly pay

#### (a) Find Tom’s pay as a function of the number of hours he worked.

Let $ P(h) $ represent Tom’s weekly pay as a function of the hours worked $ h $. We know two points: when Tom works 40 hours, his pay is \$376, and when he works 50 hours, his pay is \$390.

The function $ P(h) = ah + b $ is linear, so we need to find the slope $ a $ and the intercept $ b $.

##### Step 1: Find the slope $ a $

The slope $ a $ is the rate of change of pay with respect to hours worked:

\[
a = \frac{390 - 376}{50 - 40} = \frac{14}{10} = 1.4
\]

##### Step 2: Find the intercept $ b $

Now, use the point $ (40, 376) $ and the formula $ P(h) = ah + b $ to solve for $ b $:

\[
376 = 1.4(40) + b
\]
\[
376 = 56 + b
\]
\[
b = 376 - 56 = 320
\]

Thus, the pay function is:

\[
P(h) = 1.4h + 320
\]

#### (b) Does the pay increase as hours increase? If yes, at what rate?

Yes, the pay increases as hours increase. The rate of increase is given by the slope $ a = 1.4 $, meaning that for every additional hour worked, Tom's pay increases by \$1.40.

---

Would you like more details, or do you have any questions?

---

Here are 5 related questions for further exploration:
1. How would the function change in problem 21 if water was pumped out at a different rate?
2. In problem 22, can you find the inverse function of $ f(x) $?
3. What is the domain and range of the function $ P(h) $ in problem 23?
4. How does the slope of a linear function affect its graph?
5. How can you verify that the formula for $ f(x) $ in problem 22 is correct using the table?

**Tip**: To check if a linear equation is correct, substitute known values of $ x $ into the equation to see if they match the corresponding $ f(x) $.

21. A tank initially has 420 gallons of water. Water is getting pumped out of this tank at the rate 3 gallons/minute. (a) Write a function that relates the volume of the water to time. (b) How long will it take for the tank to be empty? 22. A table of values of a linear function is given. Find a formula for the function in the form f(x) = ax + b. x f(x) 2 5 4 8 23. Toms weekly pay is a linear function of the number of hours he worked that week. When he worked 40 hours on a week his pay was $376. When he worked 50 hours another week, the pay was $390. (a) Find Tom’s pay as a function of the number of hours he worked. (b) Does the pay increase as hours increase? If yes, at what rate?

Solve a linear equation problem involving water being pumped from a tank, write the function f(x) from a table of values, and calculate weekly pay based on hours worked. Learn key algebra concepts like rates of change and slope-intercept form. Suitable for Grades 8-10.

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