Apply Volume Formulas - 8th Grade Math
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What is the volume of a cone with diameter $8$ and height $9$ in terms of $\pi$?
What is the volume of a cone with diameter $8$ and height $9$ in terms of $\pi$?
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$48\pi$. Diameter 8 gives $r=4$; $V=\frac{1}{3}\pi(4^2)(9)=48\pi$.
$48\pi$. Diameter 8 gives $r=4$; $V=\frac{1}{3}\pi(4^2)(9)=48\pi$.
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A cylinder has $r=2$. If the height doubles, how does the volume change?
A cylinder has $r=2$. If the height doubles, how does the volume change?
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Volume doubles. Volume $V=\pi r^2h$ is proportional to height.
Volume doubles. Volume $V=\pi r^2h$ is proportional to height.
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A sphere’s radius is tripled. By what factor does the volume change?
A sphere’s radius is tripled. By what factor does the volume change?
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$27$. Volume scales with $r^3$: if $r$ triples, $V$ increases by $3^3=27$.
$27$. Volume scales with $r^3$: if $r$ triples, $V$ increases by $3^3=27$.
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Find and correct the error: A student wrote cylinder volume as $V=2\pi rh$.
Find and correct the error: A student wrote cylinder volume as $V=2\pi rh$.
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Correct: $V=\pi r^2h$. Student confused cylinder volume with lateral surface area formula.
Correct: $V=\pi r^2h$. Student confused cylinder volume with lateral surface area formula.
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State the formula for the volume of a cone with radius $r$ and height $h$.
State the formula for the volume of a cone with radius $r$ and height $h$.
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$V=\frac{1}{3}\pi r^2h$. Cone volume is one-third of cylinder volume with same base and height.
$V=\frac{1}{3}\pi r^2h$. Cone volume is one-third of cylinder volume with same base and height.
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State the formula for the volume of a cylinder with radius $r$ and height $h$.
State the formula for the volume of a cylinder with radius $r$ and height $h$.
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$V=\pi r^2h$. Base area $\pi r^2$ times height gives cylinder volume.
$V=\pi r^2h$. Base area $\pi r^2$ times height gives cylinder volume.
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What is the height $h$ of a cone with $V=48\pi$ and $r=6$?
What is the height $h$ of a cone with $V=48\pi$ and $r=6$?
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$4$. Solve $48\pi=\frac{1}{3}\pi(6^2)h$: $48\pi=12\pi h$, so $h=4$.
$4$. Solve $48\pi=\frac{1}{3}\pi(6^2)h$: $48\pi=12\pi h$, so $h=4$.
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What is the height $h$ of a cylinder with $V=64\pi$ and $r=4$?
What is the height $h$ of a cylinder with $V=64\pi$ and $r=4$?
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$4$. Solve $64\pi=\pi(4^2)h$ for $h$: $64\pi=16\pi h$, so $h=4$.
$4$. Solve $64\pi=\pi(4^2)h$ for $h$: $64\pi=16\pi h$, so $h=4$.
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If a cone and cylinder have the same $r$ and $h$, what fraction of the cylinder's volume is the cone's volume?
If a cone and cylinder have the same $r$ and $h$, what fraction of the cylinder's volume is the cone's volume?
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$\frac{1}{3}$. Cone volume formula has factor $\frac{1}{3}$ compared to cylinder.
$\frac{1}{3}$. Cone volume formula has factor $\frac{1}{3}$ compared to cylinder.
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What is the volume of a sphere with diameter $12$ in terms of $\pi$?
What is the volume of a sphere with diameter $12$ in terms of $\pi$?
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$288\pi$. Diameter 12 gives $r=6$; $V=\frac{4}{3}\pi(6^3)=288\pi$.
$288\pi$. Diameter 12 gives $r=6$; $V=\frac{4}{3}\pi(6^3)=288\pi$.
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What is the radius of a sphere with diameter $12$?
What is the radius of a sphere with diameter $12$?
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$6$. Radius is half the diameter: $12÷2=6$.
$6$. Radius is half the diameter: $12÷2=6$.
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What is the volume of a cylinder with diameter $10$ and height $4$ in terms of $\pi$?
What is the volume of a cylinder with diameter $10$ and height $4$ in terms of $\pi$?
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$100\pi$. Diameter 10 gives $r=5$; $V=\pi(5^2)(4)=100\pi$.
$100\pi$. Diameter 10 gives $r=5$; $V=\pi(5^2)(4)=100\pi$.
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Identify the missing factor: A cone has volume $V=_\pi r^2h$.
Identify the missing factor: A cone has volume $V=_\pi r^2h$.
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$\frac{1}{3}$. Cone formula has factor $\frac{1}{3}$ before $\pi r^2h$.
$\frac{1}{3}$. Cone formula has factor $\frac{1}{3}$ before $\pi r^2h$.
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What is the volume of a sphere with $r=3$ in terms of $\pi$?
What is the volume of a sphere with $r=3$ in terms of $\pi$?
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$36\pi$. Apply $V=\frac{4}{3}\pi r^3$ with $r=3$: $\frac{4}{3}\pi(27)=36\pi$.
$36\pi$. Apply $V=\frac{4}{3}\pi r^3$ with $r=3$: $\frac{4}{3}\pi(27)=36\pi$.
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What is the volume of a cone with $r=3$ and $h=5$ in terms of $\pi$?
What is the volume of a cone with $r=3$ and $h=5$ in terms of $\pi$?
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$15\pi$. Apply $V=\frac{1}{3}\pi r^2h$ with $r=3$, $h=5$: $\frac{1}{3}\pi(9)(5)=15\pi$.
$15\pi$. Apply $V=\frac{1}{3}\pi r^2h$ with $r=3$, $h=5$: $\frac{1}{3}\pi(9)(5)=15\pi$.
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What is the volume of a cylinder with $r=3$ and $h=5$ in terms of $\pi$?
What is the volume of a cylinder with $r=3$ and $h=5$ in terms of $\pi$?
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$45\pi$. Apply $V=\pi r^2h$ with $r=3$, $h=5$: $\pi(3^2)(5)=45\pi$.
$45\pi$. Apply $V=\pi r^2h$ with $r=3$, $h=5$: $\pi(3^2)(5)=45\pi$.
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State the formula for the volume of a sphere with radius $r$.
State the formula for the volume of a sphere with radius $r$.
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$V=\frac{4}{3}\pi r^3$. Sphere volume uses radius cubed with coefficient $\frac{4}{3}\pi$.
$V=\frac{4}{3}\pi r^3$. Sphere volume uses radius cubed with coefficient $\frac{4}{3}\pi$.
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What is the radius $r$ of a cylinder with $V=36\pi$ and $h=4$?
What is the radius $r$ of a cylinder with $V=36\pi$ and $h=4$?
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$3$. Solve $36\pi=\pi r^2(4)$ for $r$: $r^2=9$, so $r=3$.
$3$. Solve $36\pi=\pi r^2(4)$ for $r$: $r^2=9$, so $r=3$.
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What is the radius $r$ of a sphere with $V=\frac{4}{3}\pi(2^3)$?
What is the radius $r$ of a sphere with $V=\frac{4}{3}\pi(2^3)$?
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$2$. Given expression shows $r^3=8$, so $r=2$.
$2$. Given expression shows $r^3=8$, so $r=2$.
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Choose the correct volume formula for a sphere: $\pi r^2h$, $\frac{4}{3}\pi r^3$, or $\frac{1}{3}\pi r^2h$.
Choose the correct volume formula for a sphere: $\pi r^2h$, $\frac{4}{3}\pi r^3$, or $\frac{1}{3}\pi r^2h$.
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$V=\frac{4}{3}\pi r^3$. Sphere formula has $r^3$, not $r^2h$.
$V=\frac{4}{3}\pi r^3$. Sphere formula has $r^3$, not $r^2h$.
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Find the volume of a cone with diameter $12$ and height $4$ in terms of $\pi$.
Find the volume of a cone with diameter $12$ and height $4$ in terms of $\pi$.
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$48\pi$. Diameter $12$ gives $r=6$, so $V=\frac{1}{3}\pi(6)^2(4)$.
$48\pi$. Diameter $12$ gives $r=6$, so $V=\frac{1}{3}\pi(6)^2(4)$.
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Find the volume of a cylinder with diameter $10$ and height $3$ in terms of $\pi$.
Find the volume of a cylinder with diameter $10$ and height $3$ in terms of $\pi$.
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$75\pi$. Diameter $10$ gives $r=5$, so $V=\pi(5)^2(3)$.
$75\pi$. Diameter $10$ gives $r=5$, so $V=\pi(5)^2(3)$.
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Find and correct the error: A student wrote the sphere volume as $V=4\pi r^3$.
Find and correct the error: A student wrote the sphere volume as $V=4\pi r^3$.
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Correct: $V=\frac{4}{3}\pi r^3$. Missing the fraction $\frac{1}{3}$ in the coefficient.
Correct: $V=\frac{4}{3}\pi r^3$. Missing the fraction $\frac{1}{3}$ in the coefficient.
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Which option is the correct volume formula for a cone: $\pi r^2h$ or $\frac{1}{3}\pi r^2h$?
Which option is the correct volume formula for a cone: $\pi r^2h$ or $\frac{1}{3}\pi r^2h$?
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$V=\frac{1}{3}\pi r^2h$. Cone uses one-third of base times height.
$V=\frac{1}{3}\pi r^2h$. Cone uses one-third of base times height.
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Which option is the correct volume formula for a cylinder: $\pi r^2h$ or $\frac{1}{3}\pi r^2h$?
Which option is the correct volume formula for a cylinder: $\pi r^2h$ or $\frac{1}{3}\pi r^2h$?
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$V=\pi r^2h$. Cylinder uses full base times height, not one-third.
$V=\pi r^2h$. Cylinder uses full base times height, not one-third.
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Find the radius $r$ of a sphere with volume $V=\frac{4}{3}\pi\cdot 27$.
Find the radius $r$ of a sphere with volume $V=\frac{4}{3}\pi\cdot 27$.
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$r=3$. Since $V=36\pi=\frac{4}{3}\pi(27)$, and $27=3^3$, so $r=3$.
$r=3$. Since $V=36\pi=\frac{4}{3}\pi(27)$, and $27=3^3$, so $r=3$.
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Find the height $h$ of a cone with volume $V=12\pi$ and radius $r=3$.
Find the height $h$ of a cone with volume $V=12\pi$ and radius $r=3$.
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$h=4$. From $12\pi=\frac{1}{3}\pi(3)^2h$, solve $12\pi=3\pi h$ for $h$.
$h=4$. From $12\pi=\frac{1}{3}\pi(3)^2h$, solve $12\pi=3\pi h$ for $h$.
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Find the height $h$ of a cylinder with volume $V=50\pi$ and radius $r=5$.
Find the height $h$ of a cylinder with volume $V=50\pi$ and radius $r=5$.
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$h=2$. From $50\pi=\pi(5)^2h$, divide by $25\pi$ to get $h=2$.
$h=2$. From $50\pi=\pi(5)^2h$, divide by $25\pi$ to get $h=2$.
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Find the radius $r$ of a cylinder with volume $V=64\pi$ and height $h=16$.
Find the radius $r$ of a cylinder with volume $V=64\pi$ and height $h=16$.
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$r=2$. From $64\pi=\pi r^2(16)$, divide by $16\pi$ to get $r^2=4$.
$r=2$. From $64\pi=\pi r^2(16)$, divide by $16\pi$ to get $r^2=4$.
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Find the volume of a sphere with $r=3$ in terms of $\pi$.
Find the volume of a sphere with $r=3$ in terms of $\pi$.
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$36\pi$. Apply $V=\frac{4}{3}\pi r^3=\frac{4}{3}\pi(3)^3=\frac{4}{3}\pi(27)$.
$36\pi$. Apply $V=\frac{4}{3}\pi r^3=\frac{4}{3}\pi(3)^3=\frac{4}{3}\pi(27)$.
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