QuestionFind the volume of a cylinder with altitude 12 and base radius 15, given a similar cylinder with radius 9. Options: A. $1620 \pi$ B. $972 \pi$ C. $2700 \pi$ D. $4500 \pi$ E. $1333 \pi$

Studdy Solution

STEP 1

Assumptions1. The original cylinder has an altitude of12 and a base with radius9. The similar cylinder has a base with a radius of153. The volume of a cylinder is given by the formula $V = \pi r^ h$ , where $r$ is the radius of the base and $h$ is the height (or altitude) of the cylinder4. Two cylinders are similar if the ratio of their corresponding dimensions are equal

STEP 2

First, we need to find the ratio of the radii of the two cylinders. This can be done by dividing the radius of the similar cylinder by the radius of the original cylinder.
$Ratio = \frac{Radius_{similar}}{Radius_{original}}$

STEP 3

Now, plug in the given values for the radii of the similar and original cylinders to calculate the ratio.
$Ratio = \frac{15}{9}$

STEP 4

Calculate the ratio.
$Ratio = \frac{15}{9} = \frac{}{3}$

STEP 5

Since the cylinders are similar, the ratio of their volumes is the cube of the ratio of their corresponding dimensions. Therefore, we need to cube the ratio we just found.
$Volume\, ratio = (Ratio)^3$

STEP 6

Plug in the value for the ratio to calculate the volume ratio.
$Volume\, ratio = \left(\frac{5}{3}\right)^3$

STEP 7

Calculate the volume ratio.
$Volume\, ratio = \left(\frac{5}{3}\right)^3 = \frac{125}{27}$

STEP 8

Now that we have the volume ratio, we can find the volume of the similar cylinder by multiplying the volume of the original cylinder by the volume ratio.
$Volume_{similar} = Volume_{original} \\times Volume\, ratio$

STEP 9

First, we need to calculate the volume of the original cylinder. We can do this by using the formula for the volume of a cylinder.
$Volume_{original} = \pi (Radius_{original})^2 (Altitude_{original})$

STEP 10

Plug in the given values for the radius and altitude of the original cylinder to calculate its volume.
$Volume_{original} = \pi (9)^2 (12)$

STEP 11

Calculate the volume of the original cylinder.
$Volume_{original} = \pi (9)^ () =972\pi$

STEP 12

Now, plug in the values for the volume of the original cylinder and the volume ratio to calculate the volume of the similar cylinder.
$Volume_{similar} =972\pi \\times \frac{125}{27}$

STEP 13

Calculate the volume of the similar cylinder.
$Volume_{similar} =972\pi \\times \frac{125}{27} =4500\pi$ The volume of the similar cylinder is $4500\pi$ cubic units.

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QuestionFind the volume of a cylinder with altitude 12 and base radius 15, given a similar cylinder with radius 9. Options: A. $1620 \pi$ B. $972 \pi$ C. $2700 \pi$ D. $4500 \pi$ E. $1333 \pi$

Studdy Solution

STEP 1

STEP 2

First, we need to find the ratio of the radii of the two cylinders. This can be done by dividing the radius of the similar cylinder by the radius of the original cylinder.
$Ratio = \frac{Radius_{similar}}{Radius_{original}}$

STEP 3

Now, plug in the given values for the radii of the similar and original cylinders to calculate the ratio.
$Ratio = \frac{15}{9}$

STEP 4

Calculate the ratio.
$Ratio = \frac{15}{9} = \frac{}{3}$

STEP 5

Since the cylinders are similar, the ratio of their volumes is the cube of the ratio of their corresponding dimensions. Therefore, we need to cube the ratio we just found.
$Volume\, ratio = (Ratio)^3$

STEP 6

Plug in the value for the ratio to calculate the volume ratio.
$Volume\, ratio = \left(\frac{5}{3}\right)^3$

STEP 7

Calculate the volume ratio.
$Volume\, ratio = \left(\frac{5}{3}\right)^3 = \frac{125}{27}$

STEP 8

Now that we have the volume ratio, we can find the volume of the similar cylinder by multiplying the volume of the original cylinder by the volume ratio.
$Volume_{similar} = Volume_{original} \\times Volume\, ratio$

STEP 9

First, we need to calculate the volume of the original cylinder. We can do this by using the formula for the volume of a cylinder.
$Volume_{original} = \pi (Radius_{original})^2 (Altitude_{original})$

STEP 10

Plug in the given values for the radius and altitude of the original cylinder to calculate its volume.
$Volume_{original} = \pi (9)^2 (12)$

STEP 11

Calculate the volume of the original cylinder.
$Volume_{original} = \pi (9)^ () =972\pi$

STEP 12

Now, plug in the values for the volume of the original cylinder and the volume ratio to calculate the volume of the similar cylinder.
$Volume_{similar} =972\pi \\times \frac{125}{27}$

STEP 13

Calculate the volume of the similar cylinder.
$Volume_{similar} =972\pi \\times \frac{125}{27} =4500\pi$ The volume of the similar cylinder is $4500\pi$ cubic units.

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QuestionFind the volume of a cylinder with altitude 12 and base radius 15, given a similar cylinder with radius 9. Options: A. 1620π1620 \pi1620π B. 972π972 \pi972π C. 2700π2700 \pi2700π D. 4500π4500 \pi4500π E. 1333π1333 \pi1333π

Studdy Solution

STEP 1

STEP 2

First, we need to find the ratio of the radii of the two cylinders. This can be done by dividing the radius of the similar cylinder by the radius of the original cylinder.Ratio=RadiussimilarRadiusoriginalRatio = \frac{Radius_{similar}}{Radius_{original}}Ratio=Radiusoriginal​Radiussimilar​​

STEP 3

Now, plug in the given values for the radii of the similar and original cylinders to calculate the ratio.Ratio=159Ratio = \frac{15}{9}Ratio=915​

STEP 4

Calculate the ratio.Ratio=159=3Ratio = \frac{15}{9} = \frac{}{3}Ratio=915​=3​

STEP 5

Since the cylinders are similar, the ratio of their volumes is the cube of the ratio of their corresponding dimensions. Therefore, we need to cube the ratio we just found.Volume ratio=(Ratio)3Volume\, ratio = (Ratio)^3Volumeratio=(Ratio)3

STEP 6

Plug in the value for the ratio to calculate the volume ratio.Volume ratio=(53)3Volume\, ratio = \left(\frac{5}{3}\right)^3Volumeratio=(35​)3

STEP 7

Calculate the volume ratio.Volume ratio=(53)3=12527Volume\, ratio = \left(\frac{5}{3}\right)^3 = \frac{125}{27}Volumeratio=(35​)3=27125​

STEP 8

STEP 9

STEP 10

Plug in the given values for the radius and altitude of the original cylinder to calculate its volume.Volumeoriginal=π(9)2(12)Volume_{original} = \pi (9)^2 (12)Volumeoriginal​=π(9)2(12)

STEP 11

Calculate the volume of the original cylinder.Volumeoriginal=π(9)()=972πVolume_{original} = \pi (9)^ () =972\piVolumeoriginal​=π(9)()=972π

STEP 12

Now, plug in the values for the volume of the original cylinder and the volume ratio to calculate the volume of the similar cylinder.Volumesimilar=972πtimes12527Volume_{similar} =972\pi \\times \frac{125}{27}Volumesimilar​=972πtimes27125​

STEP 13

Calculate the volume of the similar cylinder.Volumesimilar=972πtimes12527=4500πVolume_{similar} =972\pi \\times \frac{125}{27} =4500\piVolumesimilar​=972πtimes27125​=4500πThe volume of the similar cylinder is 4500π4500\pi4500π cubic units.

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QuestionFind the volume of a cylinder with altitude 12 and base radius 15, given a similar cylinder with radius 9. Options: A. $1620 \pi$ B. $972 \pi$ C. $2700 \pi$ D. $4500 \pi$ E. $1333 \pi$

First, we need to find the ratio of the radii of the two cylinders. This can be done by dividing the radius of the similar cylinder by the radius of the original cylinder.
$Ratio = \frac{Radius_{similar}}{Radius_{original}}$

Now, plug in the given values for the radii of the similar and original cylinders to calculate the ratio.
$Ratio = \frac{15}{9}$

Calculate the ratio.
$Ratio = \frac{15}{9} = \frac{}{3}$

Since the cylinders are similar, the ratio of their volumes is the cube of the ratio of their corresponding dimensions. Therefore, we need to cube the ratio we just found.
$Volume\, ratio = (Ratio)^3$

Plug in the value for the ratio to calculate the volume ratio.
$Volume\, ratio = \left(\frac{5}{3}\right)^3$

Calculate the volume ratio.
$Volume\, ratio = \left(\frac{5}{3}\right)^3 = \frac{125}{27}$

Plug in the given values for the radius and altitude of the original cylinder to calculate its volume.
$Volume_{original} = \pi (9)^2 (12)$

Calculate the volume of the original cylinder.
$Volume_{original} = \pi (9)^ () =972\pi$

Now, plug in the values for the volume of the original cylinder and the volume ratio to calculate the volume of the similar cylinder.
$Volume_{similar} =972\pi \\times \frac{125}{27}$

Calculate the volume of the similar cylinder.
$Volume_{similar} =972\pi \\times \frac{125}{27} =4500\pi$ The volume of the similar cylinder is $4500\pi$ cubic units.