Mathematical Modeling - Pre Algebra

1. In a model where $n$ counts people, which values of $n$ make sense?

   Whole numbers only: $0, 1, 2, 3$ and so on. A count of people cannot be negative or fractional.

2. A phone plan charges 25 dollars plus 0.10 dollars per minute over 500 minutes. What is the bill for 625 minutes?

   37.50 dollars. Only 125 minutes are charged: $25 + 0.10(125) = 37.5$.

3. The temperature is $60^\circ$F and falls $2^\circ$F each hour. What equation models the temperature $T$ after $h$ hours?

   $T = 60 - 2h$.

4. A car travels 150 miles in 2.5 hours. What is its average speed?

   60 miles per hour, since $150 \div 2.5 = 60$.

5. A tank drains at 12 gallons per minute while water is added at 8 gallons per minute. What is the net rate of change?

   A loss of 4 gallons per minute, since $8 - 12 = -4$.

6. After computing a result from a model, what should you do before reporting it?

   Interpret it in the context and check that it makes sense, such as being a whole number or a positive amount where required.

7. A 500 gallon tank loses a net 4 gallons per minute. How many gallons remain after 45 minutes?

   320 gallons, since $500 - 4(45) = 500 - 180$.

8. A model shows that 4.3 buses are needed to carry a group. How many buses should be ordered?

   5 buses. A partial bus is not possible, and 4 buses would leave people without seats.

9. Two people walk in opposite directions from the same point at 3 and 4 miles per hour. How far apart are they after 2 hours?

   14 miles. They separate at $3 + 4 = 7$ miles per hour, and $7 \times 2 = 14$.

10. A recipe uses 3 cups of flour for every 2 cups of sugar. How much flour goes with 10 cups of sugar?

    15 cups, since $\frac{3}{2} \times 10 = 15$.

11. A rectangle has perimeter 60 ft, and its length $l$ is 8 ft more than twice its width $w$. What system models this?

    $2l + 2w = 60$ and $l = 2w + 8$.

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