Applying Density in Modeling Situations - Geometry
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What is the volume density if $300$ units fill a cylinder of volume $60 (\text{ft})^3$?
What is the volume density if $300$ units fill a cylinder of volume $60 (\text{ft})^3$?
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$5 (\text{units/ft})^3$. Density = $\frac{300}{60} = 5$ units per cubic foot.
$5 (\text{units/ft})^3$. Density = $\frac{300}{60} = 5$ units per cubic foot.
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What is the definition of volume density in a modeling situation?
What is the definition of volume density in a modeling situation?
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Amount per unit volume, such as $\frac{\text{quantity}}{\text{unit}^3}$. Ratio of total quantity to the volume it occupies.
Amount per unit volume, such as $\frac{\text{quantity}}{\text{unit}^3}$. Ratio of total quantity to the volume it occupies.
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What is $3\text{ BTU/ft}^3$ expressed in BTU per $
\text{in}^3$?
What is $3\text{ BTU/ft}^3$ expressed in BTU per $ \text{in}^3$?
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$\frac{1}{576}\text{ BTU/in}^3$. $\frac{3}{1728} = \frac{1}{576}$ BTU per cubic inch.
$\frac{1}{576}\text{ BTU/in}^3$. $\frac{3}{1728} = \frac{1}{576}$ BTU per cubic inch.
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Which expression correctly converts $2\text{ people/ft}^2$ to people per $\text{in}^2$?
Which expression correctly converts $2\text{ people/ft}^2$ to people per $\text{in}^2$?
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$\frac{2}{144}\text{ people/in}^2$. Convert using $1 \text{ ft}^2 = 144 \text{ in}^2$.
$\frac{2}{144}\text{ people/in}^2$. Convert using $1 \text{ ft}^2 = 144 \text{ in}^2$.
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What is $2\text{ people/ft}^2$ expressed in people per $ \text{in}^2$?
What is $2\text{ people/ft}^2$ expressed in people per $ \text{in}^2$?
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$\frac{1}{72}\text{ people/in}^2$. $\frac{2}{144} = \frac{1}{72}$ people per square inch.
$\frac{1}{72}\text{ people/in}^2$. $\frac{2}{144} = \frac{1}{72}$ people per square inch.
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Which expression correctly converts $3\text{ BTU/ft}^3$ to BTU per $\text{in}^3$?
Which expression correctly converts $3\text{ BTU/ft}^3$ to BTU per $\text{in}^3$?
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$\frac{3}{1728}\text{ BTU/in}^3$. Convert using $1 \text{ ft}^3 = 1728 \text{ in}^3$.
$\frac{3}{1728}\text{ BTU/in}^3$. Convert using $1 \text{ ft}^3 = 1728 \text{ in}^3$.
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What unit type should area density have (in terms of powers of length)?
What unit type should area density have (in terms of powers of length)?
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Units of the form $ \frac{\text{quantity}}{\text{length}^2} $. Area has square units, so density uses length squared in denominator.
Units of the form $ \frac{\text{quantity}}{\text{length}^2} $. Area has square units, so density uses length squared in denominator.
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What unit type should volume density have (in terms of powers of length)?
What unit type should volume density have (in terms of powers of length)?
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Units of the form $\frac{\text{quantity}}{\text{length}^3}$. Volume has cubic units, so density uses length cubed in denominator.
Units of the form $\frac{\text{quantity}}{\text{length}^3}$. Volume has cubic units, so density uses length cubed in denominator.
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What is the definition of area density in a modeling situation?
What is the definition of area density in a modeling situation?
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Amount per unit area, such as $ \frac{\text{quantity}}{\text{unit}^2} $. Ratio of total quantity to the area it occupies.
Amount per unit area, such as $ \frac{\text{quantity}}{\text{unit}^2} $. Ratio of total quantity to the area it occupies.
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What is the formula for total amount $Q$ given volume density $D$ and volume $V$?
What is the formula for total amount $Q$ given volume density $D$ and volume $V$?
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$Q=DV$. Multiply density by volume to get total amount.
$Q=DV$. Multiply density by volume to get total amount.
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What is the formula for total amount $Q$ given area density $D$ and area $A$?
What is the formula for total amount $Q$ given area density $D$ and area $A$?
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$Q=DA$. Multiply density by area to get total amount.
$Q=DA$. Multiply density by area to get total amount.
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What is the formula for density using volume $V$ and total amount $Q$?
What is the formula for density using volume $V$ and total amount $Q$?
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$D=\frac{Q}{V}$. Density equals quantity divided by volume.
$D=\frac{Q}{V}$. Density equals quantity divided by volume.
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What is the formula for density using area $A$ and total amount $Q$?
What is the formula for density using area $A$ and total amount $Q$?
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$D=\frac{Q}{A}$. Density equals quantity divided by area.
$D=\frac{Q}{A}$. Density equals quantity divided by area.
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Identify the correct density unit: "g/cm$^2$" or "g/cm$^3$" for volume density.
Identify the correct density unit: "g/cm$^2$" or "g/cm$^3$" for volume density.
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"g/cm$^3$". Volume density requires cubic units in the denominator.
"g/cm$^3$". Volume density requires cubic units in the denominator.
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Identify the correct density unit: "g/cm$^2$" or "g/cm$^3$" for area density.
Identify the correct density unit: "g/cm$^2$" or "g/cm$^3$" for area density.
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"g/cm$^2$". Area density requires square units in the denominator.
"g/cm$^2$". Area density requires square units in the denominator.
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What is the area of a rectangle with sides $8\text{ m}$ and $3\text{ m}$?
What is the area of a rectangle with sides $8\text{ m}$ and $3\text{ m}$?
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$24\text{ m}^2$. Area = length × width = $8 × 3$.
$24\text{ m}^2$. Area = length × width = $8 × 3$.
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What is the volume of a rectangular prism $2\text{ m}\times 3\text{ m}\times 4\text{ m}$?
What is the volume of a rectangular prism $2\text{ m}\times 3\text{ m}\times 4\text{ m}$?
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$24\text{ m}^3$. Volume = length × width × height = $2 × 3 × 4$.
$24\text{ m}^3$. Volume = length × width × height = $2 × 3 × 4$.
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What is the area density if $120$ people live in $3\text{ mi}^2$?
What is the area density if $120$ people live in $3\text{ mi}^2$?
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$40\text{ people/mi}^2$. Density = $\frac{120}{3} = 40$ people per square mile.
$40\text{ people/mi}^2$. Density = $\frac{120}{3} = 40$ people per square mile.
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What is the area density if $450$ trees occupy $9\text{ acres}$?
What is the area density if $450$ trees occupy $9\text{ acres}$?
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$50\text{ trees/acre}$. Density = $\frac{450}{9} = 50$ trees per acre.
$50\text{ trees/acre}$. Density = $\frac{450}{9} = 50$ trees per acre.
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What is the volume density if $600\text{ BTU}$ are in $30\text{ ft}^3$?
What is the volume density if $600\text{ BTU}$ are in $30\text{ ft}^3$?
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$20\text{ BTU/ft}^3$. Density = $\frac{600}{30} = 20$ BTU per cubic foot.
$20\text{ BTU/ft}^3$. Density = $\frac{600}{30} = 20$ BTU per cubic foot.
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What is the volume density if $18\text{ g}$ of material fills $6\text{ cm}^3$?
What is the volume density if $18\text{ g}$ of material fills $6\text{ cm}^3$?
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$3\text{ g/cm}^3$. Density = $\frac{18}{6} = 3$ grams per cubic centimeter.
$3\text{ g/cm}^3$. Density = $\frac{18}{6} = 3$ grams per cubic centimeter.
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What total population $Q$ corresponds to $35\text{ people/mi}^2$ over $8\text{ mi}^2$?
What total population $Q$ corresponds to $35\text{ people/mi}^2$ over $8\text{ mi}^2$?
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$280\text{ people}$. $Q = DA = 35 × 8 = 280$ people.
$280\text{ people}$. $Q = DA = 35 × 8 = 280$ people.
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What total energy $Q$ corresponds to $12\text{ BTU/ft}^3$ in $50\text{ ft}^3$?
What total energy $Q$ corresponds to $12\text{ BTU/ft}^3$ in $50\text{ ft}^3$?
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$600\text{ BTU}$. $Q = DV = 12 × 50 = 600$ BTU.
$600\text{ BTU}$. $Q = DV = 12 × 50 = 600$ BTU.
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What area $A$ is needed for $240$ people at $30\text{ people/mi}^2$?
What area $A$ is needed for $240$ people at $30\text{ people/mi}^2$?
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$8\text{ mi}^2$. $A = \frac{Q}{D} = \frac{240}{30} = 8$ square miles.
$8\text{ mi}^2$. $A = \frac{Q}{D} = \frac{240}{30} = 8$ square miles.
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What volume $V$ is needed for $900\text{ BTU}$ at $15\text{ BTU/ft}^3$?
What volume $V$ is needed for $900\text{ BTU}$ at $15\text{ BTU/ft}^3$?
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$60\text{ ft}^3$. $V = \frac{Q}{D} = \frac{900}{15} = 60$ cubic feet.
$60\text{ ft}^3$. $V = \frac{Q}{D} = \frac{900}{15} = 60$ cubic feet.
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Identify the density if $Q=72$ and $A=9$: what is $D$ in $D=\frac{Q}{A}$?
Identify the density if $Q=72$ and $A=9$: what is $D$ in $D=\frac{Q}{A}$?
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$8$. $D = \frac{72}{9} = 8$ units per area.
$8$. $D = \frac{72}{9} = 8$ units per area.
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Identify the density if $Q=84$ and $V=7$: what is $D$ in $D=\frac{Q}{V}$?
Identify the density if $Q=84$ and $V=7$: what is $D$ in $D=\frac{Q}{V}$?
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$12$. $D = \frac{84}{7} = 12$ units per volume.
$12$. $D = \frac{84}{7} = 12$ units per volume.
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What is the area of a triangle with base $10\text{ m}$ and height $6\text{ m}$?
What is the area of a triangle with base $10\text{ m}$ and height $6\text{ m}$?
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$30\text{ m}^2$. Area = $\frac{1}{2} × 10 × 6 = 30$ square meters.
$30\text{ m}^2$. Area = $\frac{1}{2} × 10 × 6 = 30$ square meters.
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What is the area density if $150$ fish are in a pond of area $30\text{ m}^2$?
What is the area density if $150$ fish are in a pond of area $30\text{ m}^2$?
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$5\text{ fish/m}^2$. Density = $\frac{150}{30} = 5$ fish per square meter.
$5\text{ fish/m}^2$. Density = $\frac{150}{30} = 5$ fish per square meter.
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What is the volume of a cylinder with $r=2\text{ ft}$ and $h=5\text{ ft}$ using $\pi\approx 3$?
What is the volume of a cylinder with $r=2\text{ ft}$ and $h=5\text{ ft}$ using $\pi\approx 3$?
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$60\text{ ft}^3$. Volume = $\pi r^2 h = 3 × 4 × 5 = 60$ cubic feet.
$60\text{ ft}^3$. Volume = $\pi r^2 h = 3 × 4 × 5 = 60$ cubic feet.
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