QuestionFind the volume of a cylinder with radius $3a$ cm and height $7b$ cm, given a cylinder of height $5b$ cm has volume $480$ cm³.

Studdy Solution

STEP 1

Assumptions1. The volume of the first right circular cylinder is $480 cm^3$ . The height of the first right circular cylinder is $5b cm$
3. The base radius of the second right circular cylinder is $3a cm$
4. The height of the second right circular cylinder is $7b cm$
5. $a$ and $b$ are positive constants6. The formula for the volume of a right circular cylinder is $V = \pi r^ h$ , where $r$ is the radius of the base and $h$ is the height

STEP 2

First, we need to find the radius of the first cylinder. We can do this by rearranging the formula for the volume of a cylinder.
$r = \sqrt{\frac{V}{\pi h}}$

STEP 3

Now, plug in the given values for the volume and height of the first cylinder to calculate the radius.
$r = \sqrt{\frac{480 cm^3}{\pi \times5b cm}}$

STEP 4

implify the expression to find the radius in terms of $a$ and $b$ .
$r = \sqrt{\frac{480 cm^3}{\pi b cm}} = a cm$

STEP 5

Now that we have the radius of the first cylinder in terms of $a$ and $b$ , we can find the volume of the second cylinder using the formula for the volume of a cylinder.
$V = \pi r^2 h$

STEP 6

Plug in the values for the radius and height of the second cylinder to calculate the volume.
$V = \pi (3a cm)^2 (b cm)$

STEP 7

implify the expression to find the volume of the second cylinder in terms of $a$ and $b$ .
$V =63\pi a^2 b cm^3$

STEP 8

Now, substitute the value of $a$ from step4 into the expression for the volume of the second cylinder.
$V =63\pi \left(\sqrt{\frac{480 cm^3}{5\pi b cm}}\right)^2 b cm^3$

STEP 9

implify the expression to find the volume of the second cylinder.
$V =63\pi \left(\frac{480 cm^3}{5\pi b cm}\right) b cm^3 =6048 cm^3$ The volume of the second right circular cylinder is $6048 cm^3$ .

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QuestionFind the volume of a cylinder with radius $3a$ cm and height $7b$ cm, given a cylinder of height $5b$ cm has volume $480$ cm³.

Studdy Solution

STEP 1

STEP 2

First, we need to find the radius of the first cylinder. We can do this by rearranging the formula for the volume of a cylinder.
$r = \sqrt{\frac{V}{\pi h}}$

STEP 3

Now, plug in the given values for the volume and height of the first cylinder to calculate the radius.
$r = \sqrt{\frac{480 cm^3}{\pi \times5b cm}}$

STEP 4

implify the expression to find the radius in terms of $a$ and $b$ .
$r = \sqrt{\frac{480 cm^3}{\pi b cm}} = a cm$

STEP 5

Now that we have the radius of the first cylinder in terms of $a$ and $b$ , we can find the volume of the second cylinder using the formula for the volume of a cylinder.
$V = \pi r^2 h$

STEP 6

Plug in the values for the radius and height of the second cylinder to calculate the volume.
$V = \pi (3a cm)^2 (b cm)$

STEP 7

implify the expression to find the volume of the second cylinder in terms of $a$ and $b$ .
$V =63\pi a^2 b cm^3$

STEP 8

Now, substitute the value of $a$ from step4 into the expression for the volume of the second cylinder.
$V =63\pi \left(\sqrt{\frac{480 cm^3}{5\pi b cm}}\right)^2 b cm^3$

STEP 9

implify the expression to find the volume of the second cylinder.
$V =63\pi \left(\frac{480 cm^3}{5\pi b cm}\right) b cm^3 =6048 cm^3$ The volume of the second right circular cylinder is $6048 cm^3$ .

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QuestionFind the volume of a cylinder with radius 3a3a3a cm and height 7b7b7b cm, given a cylinder of height 5b5b5b cm has volume 480480480 cm³.

Studdy Solution

STEP 1

STEP 2

First, we need to find the radius of the first cylinder. We can do this by rearranging the formula for the volume of a cylinder.r=Vπhr = \sqrt{\frac{V}{\pi h}}r=πhV​​

STEP 3

Now, plug in the given values for the volume and height of the first cylinder to calculate the radius.r=480cm3π×5bcmr = \sqrt{\frac{480 cm^3}{\pi \times5b cm}}r=π×5bcm480cm3​​

STEP 4

implify the expression to find the radius in terms of aaa and bbb.r=480cm3πbcm=acmr = \sqrt{\frac{480 cm^3}{\pi b cm}} = a cmr=πbcm480cm3​​=acm

STEP 5

Now that we have the radius of the first cylinder in terms of aaa and bbb, we can find the volume of the second cylinder using the formula for the volume of a cylinder.V=πr2hV = \pi r^2 hV=πr2h

STEP 6

Plug in the values for the radius and height of the second cylinder to calculate the volume.V=π(3acm)2(bcm)V = \pi (3a cm)^2 (b cm)V=π(3acm)2(bcm)

STEP 7

implify the expression to find the volume of the second cylinder in terms of aaa and bbb.V=63πa2bcm3V =63\pi a^2 b cm^3V=63πa2bcm3

STEP 8

Now, substitute the value of aaa from step4 into the expression for the volume of the second cylinder.V=63π(480cm35πbcm)2bcm3V =63\pi \left(\sqrt{\frac{480 cm^3}{5\pi b cm}}\right)^2 b cm^3V=63π(5πbcm480cm3​​)2bcm3

STEP 9

implify the expression to find the volume of the second cylinder.V=63π(480cm35πbcm)bcm3=6048cm3V =63\pi \left(\frac{480 cm^3}{5\pi b cm}\right) b cm^3 =6048 cm^3V=63π(5πbcm480cm3​)bcm3=6048cm3The volume of the second right circular cylinder is 6048cm36048 cm^36048cm3.

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QuestionFind the volume of a cylinder with radius $3a$ cm and height $7b$ cm, given a cylinder of height $5b$ cm has volume $480$ cm³.

First, we need to find the radius of the first cylinder. We can do this by rearranging the formula for the volume of a cylinder.
$r = \sqrt{\frac{V}{\pi h}}$

Now, plug in the given values for the volume and height of the first cylinder to calculate the radius.
$r = \sqrt{\frac{480 cm^3}{\pi \times5b cm}}$

implify the expression to find the radius in terms of $a$ and $b$ .
$r = \sqrt{\frac{480 cm^3}{\pi b cm}} = a cm$

Now that we have the radius of the first cylinder in terms of $a$ and $b$ , we can find the volume of the second cylinder using the formula for the volume of a cylinder.
$V = \pi r^2 h$

Plug in the values for the radius and height of the second cylinder to calculate the volume.
$V = \pi (3a cm)^2 (b cm)$

implify the expression to find the volume of the second cylinder in terms of $a$ and $b$ .
$V =63\pi a^2 b cm^3$

Now, substitute the value of $a$ from step4 into the expression for the volume of the second cylinder.
$V =63\pi \left(\sqrt{\frac{480 cm^3}{5\pi b cm}}\right)^2 b cm^3$

implify the expression to find the volume of the second cylinder.
$V =63\pi \left(\frac{480 cm^3}{5\pi b cm}\right) b cm^3 =6048 cm^3$ The volume of the second right circular cylinder is $6048 cm^3$ .