Applying Density in Modeling Situations - Geometry

1. Which unit best matches volume density: $\text{kg/m}^2$ or $\text{kg/m}^3$?

   $\text{kg/m}^3$. Volume density has cubic units in the denominator.

2. What total population $Q$ corresponds to $35\text{ people/mi}^2$ over $8\text{ mi}^2$?

   $280\text{ people}$. $Q = DA = 35 × 8 = 280$ people.

3. Identify the correct density unit: "g/cm$^2$" or "g/cm$^3$" for volume density.

   "g/cm$^3$". Volume density requires cubic units in the denominator.

4. What is the volume density if $300$ units fill a cylinder of volume $60 (\text{ft})^3$?

   $5 (\text{units/ft})^3$. Density = $\frac{300}{60} = 5$ units per cubic foot.

5. Which expression correctly converts $3\text{ BTU/ft}^3$ to BTU per $\text{in}^3$?

   $\frac{3}{1728}\text{ BTU/in}^3$. Convert using $1 \text{ ft}^3 = 1728 \text{ in}^3$.

6. What is the area density if $450$ trees occupy $9\text{ acres}$?

   $50\text{ trees/acre}$. Density = $\frac{450}{9} = 50$ trees per acre.

7. If a map scale factor is $k=2$, how does a fixed total $Q$ affect area density?

   Density becomes $\frac{1}{4}$ of the original. Area scales by $2^2 = 4$, so density is divided by 4.

8. Identify the correct density unit: "g/cm$^2$" or "g/cm$^3$" for area density.

   "g/cm$^2$". Area density requires square units in the denominator.

9. A room has $9600\text{ BTU}$ in $320\text{ ft}^3$. What is the energy density?

   $30\text{ BTU/ft}^3$. Density = $\frac{9600}{320} = 30$ BTU per cubic foot.

10. What is the formula for total amount $Q$ given volume density $D$ and volume $V$?

    $Q=DV$. Multiply density by volume to get total amount.

11. What is $3\text{ BTU/ft}^3$ expressed in BTU per $\text{in}^3$?

    $\frac{1}{576}\text{ BTU/in}^3$. $\frac{3}{1728} = \frac{1}{576}$ BTU per cubic inch.

12. What is the area of a triangle with base $10\text{ m}$ and height $6\text{ m}$?

    $30\text{ m}^2$. Area = $\frac{1}{2} × 10 × 6 = 30$ square meters.

13. Find and correct the unit error: area density written as $\text{people/mi}^3$.

    Correct unit: $\text{people/mi}^2$. Area density needs square units, not cubic units.

14. What is the area of a rectangle with sides $8\text{ m}$ and $3\text{ m}$?

    $24\text{ m}^2$. Area = length × width = $8 × 3$.

15. A city has $80\text{ people/mi}^2$. How many people are in $0.5\text{ mi}^2$?

    $40\text{ people}$. $Q = DA = 80 × 0.5 = 40$ people.

16. What volume $V$ is needed for $900\text{ BTU}$ at $15\text{ BTU/ft}^3$?

    $60\text{ ft}^3$. $V = \frac{Q}{D} = \frac{900}{15} = 60$ cubic feet.

17. A plot has density $4\text{ plants/m}^2$. How many plants are in $15\text{ m}^2$?

    $60\text{ plants}$. $Q = DA = 4 × 15 = 60$ plants.

18. What unit type should area density have (in terms of powers of length)?

    Units of the form $\frac{\text{quantity}}{\text{length}^2}$. Area has square units, so density uses length squared in denominator.

19. What is $2\text{ people/ft}^2$ expressed in people per $\text{in}^2$?

    $\frac{1}{72}\text{ people/in}^2$. $\frac{2}{144} = \frac{1}{72}$ people per square inch.

20. What is the definition of volume density in a modeling situation?

    Amount per unit volume, such as $\frac{\text{quantity}}{\text{unit}^3}$. Ratio of total quantity to the volume it occupies.

21. Find $A$ if $Q=500$ and $D=25$ in $D=\frac{Q}{A}$.

    $20$. $A = \frac{Q}{D} = \frac{500}{25} = 20$.

22. What is the formula for total amount $Q$ given area density $D$ and area $A$?

    $Q=DA$. Multiply density by area to get total amount.

23. Identify the correct interpretation of $D=\frac{Q}{V}$ in words.

    Density equals amount per unit volume. Formula represents quantity divided by volume.

24. Identify the density if $Q=72$ and $A=9$: what is $D$ in $D=\frac{Q}{A}$?

    $8$. $D = \frac{72}{9} = 8$ units per area.

25. What is the volume density if $300$ units fill a cylinder of volume $60\text{ ft}^3$?

    $5\text{ units/ft}^3$. Density = $\frac{300}{60} = 5$ units per cubic foot.

26. What is the area density if $150$ fish are in a pond of area $30\text{m}^2$?

    $5\text{fish/m}^2$. Density = $\frac{150}{30} = 5$ fish per square meter.

27. What is the volume of a cylinder with $r=2\text{ ft}$ and $h=5\text{ ft}$ using $\pi\approx 3$?

    $60\text{ ft}^3$. Volume = $\pi r^2 h = 3 × 4 × 5 = 60$ cubic feet.

28. A park has $300$ visitors over $12 \text{ ha}$. What is the visitor density?

    $25 \text{ visitors/ha}$. Density = $\frac{300}{12} = 25$ visitors per hectare.

29. If a solid is scaled by factor $k$, how does its volume scale?

    Volume scales by $k^3$. Linear scaling factor $k$ gives volume scaling $k^3$.

30. Identify the missing variable: $Q=DV$ solves for what when you compute $\frac{Q}{D}$?

    $V$. Rearranging $Q = DV$ gives $V = \frac{Q}{D}$.

31. What is the definition of area density in a modeling situation?

    Amount per unit area, such as $\frac{\text{quantity}}{\text{unit}^2}$. Ratio of total quantity to the area it occupies.

32. Which unit best matches area density: $\text{kg/m}^2$ or $\text{kg/m}^3$?

    $\text{kg/m}^2$. Area density has square units in the denominator.

33. What is the volume of a rectangular prism $2\text{ m}\times 3\text{ m}\times 4\text{ m}$?

    $24\text{ m}^3$. Volume = length × width × height = $2 × 3 × 4$.

34. If a model is scaled by $k=3$ with fixed total $Q$, how does volume density change?

    Density becomes $\frac{1}{27}$ of the original. Volume scales by $3^3 = 27$, so density is divided by 27.

35. Find and correct the unit error: area density written as $\text{people/mi}^3$.

    Correct unit: $\text{people/mi}^2$. Area density needs square units, not cubic units.

36. Identify the density if $Q=84$ and $V=7$: what is $D$ in $D=\frac{Q}{V}$?

    $12$. $D = \frac{84}{7} = 12$ units per volume.

37. A fuel has $25\text{ BTU/ft}^3$. How many BTU are in $2\text{ ft}^3$?

    $50\text{ BTU}$. $Q = DV = 25 × 2 = 50$ BTU.

38. What is the area density if $120$ people live in $3\text{ mi}^2$?

    $40\text{ people/mi}^2$. Density = $\frac{120}{3} = 40$ people per square mile.

39. What area $A$ is needed for $240$ people at $30\text{ people/mi}^2$?

    $8\text{ mi}^2$. $A = \frac{Q}{D} = \frac{240}{30} = 8$ square miles.

40. What is the formula for density using area $A$ and total amount $Q$?

    $D=\frac{Q}{A}$. Density equals quantity divided by area.

41. What total energy $Q$ corresponds to $12\text{ BTU/ft}^3$ in $50\text{ ft}^3$?

    $600\text{ BTU}$. $Q = DV = 12 × 50 = 600$ BTU.

42. What is the volume density if $18\text{ g}$ of material fills $6\text{ cm}^3$?

    $3\text{ g/cm}^3$. Density = $\frac{18}{6} = 3$ grams per cubic centimeter.

43. Which expression correctly converts $2\text{ people/ft}^2$ to people per $\text{in}^2$?

    $\frac{2}{144}\text{ people/in}^2$. Convert using $1 \text{ ft}^2 = 144 \text{ in}^2$.

44. Find and correct the unit error: volume density written as $\text{BTU/ft}^2$.

    Correct unit: $\text{BTU/ft}^3$. Volume density needs cubic units, not square units.

45. If a region is scaled by factor $k$, how does its area scale?

    Area scales by $k^2$. Linear scaling factor $k$ gives area scaling $k^2$.

46. What is the volume density if $600\text{ BTU}$ are in $30\text{ ft}^3$?

    $20\text{ BTU/ft}^3$. Density = $\frac{600}{30} = 20$ BTU per cubic foot.

47. A container has density $7\text{ mg/cm}^3$. How many mg are in $10\text{ cm}^3$?

    $70\text{ mg}$. $Q = DV = 7 × 10 = 70$ milligrams.

48. What does the unit "persons per square mile" represent in modeling?

    Area density: persons per $\text{mi}^2$. Population distributed over square miles measures area density.

49. What happens to area density if the same total $Q$ is spread over $2$ times the area?

    It is divided by $2$. Same quantity over double area gives half density.

50. Find $V$ if $Q=144$ and $D=6$ in $D=\frac{Q}{V}$.

    $24$. $V = \frac{Q}{D} = \frac{144}{6} = 24$.

51. What happens to volume density if the same total $Q$ is spread over $3$ times the volume?

    It is divided by $3$. Same quantity over triple volume gives one-third density.

52. What unit type should volume density have (in terms of powers of length)?

    Units of the form $\frac{\text{quantity}}{\text{length}^3}$. Volume has cubic units, so density uses length cubed in denominator.

53. What is the formula for density using volume $V$ and total amount $Q$?

    $D=\frac{Q}{V}$. Density equals quantity divided by volume.

54. A park has $300$ visitors over $12\text{ ha}$. What is the visitor density?

    $25\text{ visitors/ha}$. Density = $\frac{300}{12} = 25$ visitors per hectare.

55. What is $2\text{ people/ft}^2$ expressed in people per $\text{in}^2$?

    $\frac{1}{72}\text{ people/in}^2$. $\frac{2}{144} = \frac{1}{72}$ people per square inch.

56. Which expression correctly converts $2\text{ people/ft}^2$ to people per $\text{in}^2$?

    $\frac{2}{144}\text{ people/in}^2$. Convert using $1 \text{ ft}^2 = 144 \text{ in}^2$.

57. What does the unit "BTU per cubic foot" represent in modeling?

    Volume density: BTU per $\text{ft}^3$. Energy distributed over cubic feet measures volume density.

58. Identify the correct interpretation of $D=\frac{Q}{A}$ in words.

    Density equals amount per unit area. Formula represents quantity divided by area.

59. Identify the missing variable: $Q=DA$ solves for what when you compute $\frac{Q}{D}$?

    $A$. Rearranging $Q = DA$ gives $A = \frac{Q}{D}$.

60. What is the area density if $150$ fish are in a pond of area $30\text{ m}^2$?

    $5\text{ fish/m}^2$. Density = $\frac{150}{30} = 5$ fish per square meter.

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