Linear Function Modeling - Pre Algebra

1. A tank holds 8,000 gallons at the start and 6,800 gallons 2 hours later, draining at a constant rate. What is the rate?

   600 gallons per hour. The tank lost $8000 - 6800 = 1200$ gallons in 2 hours.

2. A car travels at a constant 55 miles per hour. What function gives the distance $d$ in miles after $t$ hours?

   $d = 55t$. Distance equals rate times time.

3. A 30 cm candle burns down 2 cm per hour. What function gives its height $h$ in centimeters after $t$ hours?

   $h = 30 - 2t$. The height starts at 30 and decreases by 2 each hour.

4. In the cost model $C = 8x + 150$, which number is the fixed cost?

   150. It does not depend on $x$, the number of items made.

5. Making $x$ shirts costs $C = 8x + 140$ dollars, and each shirt sells for 15 dollars. What function gives the profit $P$?

   $P = 7x - 140$. Profit is revenue $15x$ minus cost $8x + 140$.

6. Plan A costs 25 dollars plus 0.05 dollars per text. Plan B costs 35 dollars with unlimited texts. At how many texts do the plans cost the same?

   200 texts. Solve $25 + 0.05t = 35$: $0.05t = 10$, so $t = 200$.

7. A gym charges a 40 dollar sign-up fee plus 15 dollars per month. What is the total cost for 6 months?

   130 dollars, since $40 + 15(6) = 130$.

8. A pool holds 8,000 gallons and drains at 500 gallons per hour. After how many hours is it empty?

   16 hours. Solve $8000 - 500t = 0$ to get $t = 16$.

9. The profit on $x$ shirts is $P = 7x - 140$ dollars. How many shirts must be sold to break even?

   20 shirts. Break even means $P = 0$, so $7x = 140$ and $x = 20$.

10. A pool holds 8,000 gallons and drains at 500 gallons per hour. What function gives the gallons $G$ left after $t$ hours?

    $G = 8000 - 500t$. The amount starts at 8,000 and drops 500 each hour.

11. A taxi charges 3 dollars plus 2 dollars per mile. What function gives the cost $C$ in dollars for $m$ miles?

    $C = 2m + 3$. The per-mile charge is the rate and the 3 dollars is the starting value.

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