Let's break these problems down step by step:

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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.

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22. Linear function from a table

The table of values is:

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)$$.