Solving Problems with Volume Formulas - Geometry
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What is the relationship between a cone’s volume and a same-base, same-height cylinder’s volume?
What is the relationship between a cone’s volume and a same-base, same-height cylinder’s volume?
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$V_{\text{cone}}=\frac{1}{3}V_{\text{cyl}}$. A cone always has one-third the volume of its corresponding cylinder.
$V_{\text{cone}}=\frac{1}{3}V_{\text{cyl}}$. A cone always has one-third the volume of its corresponding cylinder.
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What is the volume of a cone with diameter $10$ and height $12$?
What is the volume of a cone with diameter $10$ and height $12$?
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$100\pi$. Diameter 10 means $r = 5$; $V = \frac{1}{3}\pi(5)^2(12)$.
$100\pi$. Diameter 10 means $r = 5$; $V = \frac{1}{3}\pi(5)^2(12)$.
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What is the volume of a cylinder with $r=2$ and $h=9$?
What is the volume of a cylinder with $r=2$ and $h=9$?
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$36\pi$. $V = \pi(2)^2(9) = 36\pi$.
$36\pi$. $V = \pi(2)^2(9) = 36\pi$.
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A cylinder’s radius doubles and height stays the same. By what factor does $V$ change?
A cylinder’s radius doubles and height stays the same. By what factor does $V$ change?
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Factor of $4$. $V \propto r^2$, so doubling $r$ gives $ (2)^2 = 4 $ times volume.
Factor of $4$. $V \propto r^2$, so doubling $r$ gives $ (2)^2 = 4 $ times volume.
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What is the volume of a pyramid if a same-base, same-height prism has volume $150$?
What is the volume of a pyramid if a same-base, same-height prism has volume $150$?
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$50$. Pyramid volume is one-third of prism volume.
$50$. Pyramid volume is one-third of prism volume.
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A cone has volume $25\pi$, radius $r=5$. What is the height $h$?
A cone has volume $25\pi$, radius $r=5$. What is the height $h$?
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$h=3$. Solve $25\pi = \frac{1}{3}\pi(5)^2 h$ for $h$.
$h=3$. Solve $25\pi = \frac{1}{3}\pi(5)^2 h$ for $h$.
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A cylinder has volume $200\pi$ and height $h=8$. What is the radius $r$?
A cylinder has volume $200\pi$ and height $h=8$. What is the radius $r$?
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$r=5$. Solve $200\pi = \pi r^2(8)$ for $r$.
$r=5$. Solve $200\pi = \pi r^2(8)$ for $r$.
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A sphere has radius $r=5$. What is its volume?
A sphere has radius $r=5$. What is its volume?
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$\frac{500}{3}\pi$. $V = \frac{4}{3}\pi(5)^3 = \frac{500\pi}{3}$.
$\frac{500}{3}\pi$. $V = \frac{4}{3}\pi(5)^3 = \frac{500\pi}{3}$.
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What is the volume of a cylinder with $r=6$ and $h=4$?
What is the volume of a cylinder with $r=6$ and $h=4$?
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$144\pi$. $V = \pi(6)^2(4) = 144\pi$.
$144\pi$. $V = \pi(6)^2(4) = 144\pi$.
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What is the volume of a cone with $r=6$ and $h=4$?
What is the volume of a cone with $r=6$ and $h=4$?
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$48\pi$. $V = \frac{1}{3}\pi(6)^2(4) = \frac{144\pi}{3}$.
$48\pi$. $V = \frac{1}{3}\pi(6)^2(4) = \frac{144\pi}{3}$.
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What is the volume of a cylinder with $r=1.5$ and $h=8$?
What is the volume of a cylinder with $r=1.5$ and $h=8$?
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$18\pi$. $V = \pi(1.5)^2(8) = \pi \cdot 2.25 \cdot 8$.
$18\pi$. $V = \pi(1.5)^2(8) = \pi \cdot 2.25 \cdot 8$.
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What is the volume of a sphere with diameter $8$?
What is the volume of a sphere with diameter $8$?
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$\frac{256}{3}\pi$. Diameter 8 means $r = 4$; $V = \frac{4}{3}\pi(4)^3$.
$\frac{256}{3}\pi$. Diameter 8 means $r = 4$; $V = \frac{4}{3}\pi(4)^3$.
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Identify the error: using $V=\frac{1}{2}Bh$ for a pyramid. What is the correct coefficient?
Identify the error: using $V=\frac{1}{2}Bh$ for a pyramid. What is the correct coefficient?
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Coefficient $\frac{1}{3}$, so $V=\frac{1}{3}Bh$. Pyramid volume uses coefficient $\frac{1}{3}$, not $\frac{1}{2}$.
Coefficient $\frac{1}{3}$, so $V=\frac{1}{3}Bh$. Pyramid volume uses coefficient $\frac{1}{3}$, not $\frac{1}{2}$.
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Identify the error: using $V=4\pi r^2$ for a sphere. What is the correct volume formula?
Identify the error: using $V=4\pi r^2$ for a sphere. What is the correct volume formula?
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$V=\frac{4}{3}\pi r^3$. Confused with surface area formula; volume uses $\frac{4}{3}\pi r^3$.
$V=\frac{4}{3}\pi r^3$. Confused with surface area formula; volume uses $\frac{4}{3}\pi r^3$.
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Identify the error: using $V=\pi r^2l$ for a cone, where $l$ is slant height. What should be used?
Identify the error: using $V=\pi r^2l$ for a cone, where $l$ is slant height. What should be used?
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Use $V=\frac{1}{3}\pi r^2h$ with perpendicular $h$. Cone volume formula uses perpendicular height, not slant height.
Use $V=\frac{1}{3}\pi r^2h$ with perpendicular $h$. Cone volume formula uses perpendicular height, not slant height.
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Find the radius of a sphere with volume $36\pi$.
Find the radius of a sphere with volume $36\pi$.
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$r=3$. Solve $36\pi = \frac{4}{3}\pi r^3$ for $r$.
$r=3$. Solve $36\pi = \frac{4}{3}\pi r^3$ for $r$.
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Find the radius of a sphere with volume $\frac{4}{3}\pi$.
Find the radius of a sphere with volume $\frac{4}{3}\pi$.
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$r=1$. Solve $\frac{4}{3}\pi = \frac{4}{3}\pi r^3$ for $r$.
$r=1$. Solve $\frac{4}{3}\pi = \frac{4}{3}\pi r^3$ for $r$.
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Find the radius of a cone with volume $12\pi$ and height $h=3$.
Find the radius of a cone with volume $12\pi$ and height $h=3$.
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$r=\sqrt{12}$. Solve $12\pi = \frac{1}{3}\pi r^2(3)$ for $r$.
$r=\sqrt{12}$. Solve $12\pi = \frac{1}{3}\pi r^2(3)$ for $r$.
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Find the height of a cone with volume $48\pi$ and radius $r=4$.
Find the height of a cone with volume $48\pi$ and radius $r=4$.
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$h=9$. Solve $48\pi = \frac{1}{3}\pi(4)^2 h$ for $h$.
$h=9$. Solve $48\pi = \frac{1}{3}\pi(4)^2 h$ for $h$.
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Find the radius of a cylinder with volume $64\pi$ and height $h=4$.
Find the radius of a cylinder with volume $64\pi$ and height $h=4$.
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$r=4$. Solve $64\pi = \pi r^2(4)$ for $r$.
$r=4$. Solve $64\pi = \pi r^2(4)$ for $r$.
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Find the height of a cylinder with volume $81\pi$ and radius $r=3$.
Find the height of a cylinder with volume $81\pi$ and radius $r=3$.
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$h=9$. Solve $81\pi = \pi(3)^2 h$ for $h$.
$h=9$. Solve $81\pi = \pi(3)^2 h$ for $h$.
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What is the volume of a sphere with $r=2$?
What is the volume of a sphere with $r=2$?
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$\frac{32}{3}\pi$. $V = \frac{4}{3}\pi(2)^3 = \frac{32}{3}\pi$.
$\frac{32}{3}\pi$. $V = \frac{4}{3}\pi(2)^3 = \frac{32}{3}\pi$.
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What is the volume of a cone with $r=2$ and $h=9$?
What is the volume of a cone with $r=2$ and $h=9$?
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$12\pi$. $V = \frac{1}{3}\pi(2)^2(9) = 12\pi$.
$12\pi$. $V = \frac{1}{3}\pi(2)^2(9) = 12\pi$.
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What is the volume of a cylinder with $r=2$ and $h=9$?
What is the volume of a cylinder with $r=2$ and $h=9$?
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$36\pi$. $V = \pi(2)^2(9) = 36\pi$.
$36\pi$. $V = \pi(2)^2(9) = 36\pi$.
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A cone’s radius doubles and height doubles. By what factor does $V$ change?
A cone’s radius doubles and height doubles. By what factor does $V$ change?
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Factor of $8$. $V \propto r^2h$, so $(2)^2 \cdot 2 = 8$ times volume.
Factor of $8$. $V \propto r^2h$, so $(2)^2 \cdot 2 = 8$ times volume.
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A sphere’s radius doubles. By what factor does the volume change?
A sphere’s radius doubles. By what factor does the volume change?
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Factor of $8$. $V \propto r^3$, so doubling $r$ gives $(2)^3 = 8$ times volume.
Factor of $8$. $V \propto r^3$, so doubling $r$ gives $(2)^3 = 8$ times volume.
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A cylinder’s height triples and radius stays the same. By what factor does $V$ change?
A cylinder’s height triples and radius stays the same. By what factor does $V$ change?
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Factor of $3$. $V \propto h$, so tripling $h$ gives $3$ times volume.
Factor of $3$. $V \propto h$, so tripling $h$ gives $3$ times volume.
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What is the volume of a cone if a same-base, same-height cylinder has volume $72\pi$?
What is the volume of a cone if a same-base, same-height cylinder has volume $72\pi$?
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$24\pi$. Cone volume is one-third of cylinder volume.
$24\pi$. Cone volume is one-third of cylinder volume.
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What is the volume of a cylinder with base area $B=18\pi$ and height $h=7$?
What is the volume of a cylinder with base area $B=18\pi$ and height $h=7$?
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$126\pi$. $V = 18\pi \cdot 7$.
$126\pi$. $V = 18\pi \cdot 7$.
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What is the volume of a cone with base area $B=27\pi$ and height $h=6$?
What is the volume of a cone with base area $B=27\pi$ and height $h=6$?
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$54\pi$. $V = \frac{1}{3} \cdot 27\pi \cdot 6 = \frac{162\pi}{3}$.
$54\pi$. $V = \frac{1}{3} \cdot 27\pi \cdot 6 = \frac{162\pi}{3}$.
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