QuestionFind the volume of a cylinder 3.769 mm high and 0.85 ft wide in $\mathrm{cm}^{3}$ using $V=\pi r^{2} h$ .

Studdy Solution

STEP 1

Assumptions1. The height of the cylinder is3.769 millimeters. The diameter of the cylinder is0.85 feet3. The volume of a cylinder is given by $V=\pi r^{} h$
4. There are exactly $.54 \mathrm{~cm} /$ inch and $12 \mathrm{in} / \mathrm{ft}$

STEP 2

First, we need to convert the height of the cylinder from millimeters to centimeters. We know that1 cm =10 mm, so we can do this by dividing the height in millimeters by10.
$Height_{cm} = Height_{mm} /10$

STEP 3

Now, plug in the given value for the height in millimeters to calculate the height in centimeters.
$Height_{cm} =3.769 \, mm /10$

STEP 4

Calculate the height in centimeters.
$Height_{cm} =3.769 \, mm /10 =0.3769 \, cm$

STEP 5

Next, we need to convert the diameter of the cylinder from feet to inches. We know that1 ft =12 in, so we can do this by multiplying the diameter in feet by12.
$iameter_{in} = Diameter_{ft} \times12$

STEP 6

Now, plug in the given value for the diameter in feet to calculate the diameter in inches.
$iameter_{in} =0.85 \, ft \times12$

STEP 7

Calculate the diameter in inches.
$iameter_{in} =0.85 \, ft \times12 =10.2 \, in$

STEP 8

Next, we need to convert the diameter of the cylinder from inches to centimeters. We know that1 in =2.54 cm, so we can do this by multiplying the diameter in inches by2.54.
$iameter_{cm} = Diameter_{in} \times2.54$

STEP 9

Now, plug in the calculated value for the diameter in inches to calculate the diameter in centimeters.
$iameter_{cm} =.2 \, in \times2.54$

STEP 10

Calculate the diameter in centimeters.
$iameter_{cm} =10.2 \, in \times2.54 =25.908 \, cm$

STEP 11

The radius of the cylinder is half the diameter. So, we divide the diameter by to find the radius.
$Radius_{cm} = Diameter_{cm} /$

STEP 12

Now, plug in the calculated value for the diameter in centimeters to calculate the radius in centimeters.
$Radius_{cm} =25.908 \, cm /2$

STEP 13

Calculate the radius in centimeters.
$Radius_{cm} =25.908 \, cm /2 =12.954 \, cm$

STEP 14

Now that we have the radius and the height in the same units (centimeters), we can calculate the volume of the cylinder using the formula $V=\pi r^{2} h$ .
$Volume_{cm^3} = \pi \times (Radius_{cm})^{2} \times Height_{cm}$

STEP 15

Now, plug in the calculated values for the radius and height in centimeters to calculate the volume in cubic centimeters.
$Volume_{cm^3} = \pi \times (12.954 \, cm)^{2} \times0.3769 \, cm$

STEP 16

Calculate the volume in cubic centimeters.
$Volume_{cm^3} = \pi \times (12.954 \, cm)^{2} \times0.3769 \, cm =198. \, cm^3$ The volume of the cylinder is $198. \, cm^3$ .

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