To calculate the mass of the cylindrical pipe, we need to follow several steps. Here's a detailed, step-by-step solution:
### Step 1: Convert the Measurements to Consistent Units
First, we need to ensure all measurements are in consistent units. We'll convert the diameters from millimeters to centimeters and the length from meters to centimeters.
1 meter = 100 centimeters
1 millimeter = 0.1 centimeters
- Length of the pipe:
[tex]\[
5 \text{ meters} = 500 \text{ centimeters}
\][/tex]
- Internal diameter:
[tex]\[
28 \text{ millimeters} = 2.8 \text{ centimeters}
\][/tex]
- External diameter:
[tex]\[
42 \text{ millimeters} = 4.2 \text{ centimeters}
\][/tex]
### Step 2: Determine the Radii
Next, we need to find the internal and external radii by dividing the diameters by 2.
- Internal radius ([tex]\( r_i \)[/tex]):
[tex]\[
r_i = \frac{2.8}{2} = 1.4 \text{ centimeters}
\][/tex]
- External radius ([tex]\( r_e \)[/tex]):
[tex]\[
r_e = \frac{4.2}{2} = 2.1 \text{ centimeters}
\][/tex]
### Step 3: Calculate the Volumes of the Cylinders
We'll use the formula for the volume of a cylinder, [tex]\( V = \pi r^2 h \)[/tex], where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height (or length) of the cylinder.
- Volume of the external cylinder:
[tex]\[
V_{\text{external}} = \pi (r_e)^2 h = \pi (2.1)^2 \times 500
\][/tex]
[tex]\[
V_{\text{external}} \approx 6927.21 \text{ cubic centimeters}
\][/tex]
- Volume of the internal cylinder:
[tex]\[
V_{\text{internal}} = \pi (r_i)^2 h = \pi (1.4)^2 \times 500
\][/tex]
[tex]\[
V_{\text{internal}} \approx 3078.76 \text{ cubic centimeters}
\][/tex]
### Step 4: Calculate the Volume of the Material of the Pipe
The volume of the material making up the pipe is the difference between the volumes of the external and internal cylinders.
[tex]\[
V_{\text{material}} = V_{\text{external}} - V_{\text{internal}}
\][/tex]
[tex]\[
V_{\text{material}} = 6927.21 - 3078.76 \approx 3848.45 \text{ cubic centimeters}
\][/tex]
### Step 5: Calculate the Mass of the Pipe
The density of the material is given as [tex]\( 1.45 \text{ g/cm}^3 \)[/tex]. To find the mass, we use the formula:
[tex]\[
m = \text{density} \times \text{volume}
\][/tex]
Before using the formula, recall we need the final mass in kilograms. Since 1 gram [tex]\( = 0.001 \)[/tex] kilograms, [tex]\( 1.45 \text{ g/cm}^3 = 0.00145 \text{ kg/cm}^3 \)[/tex].
[tex]\[
m = 1.45 \text{ g/cm}^3 \times 3848.45 \text{ cm}^3
\][/tex]
To convert the density into kg/cm³,
[tex]\[
m = 0.00145 \text{ kg/cm}^3 \times 3848.45 \text{ cm}^3 \approx 5.58 \text{ kilograms}
\][/tex]
### Final Answer
The mass of the cylindrical pipe is approximately [tex]\( 5.58 \text{ kilograms} \)[/tex].
