Answer :
We are given that the artisan has a metal mass of
[tex]\[
63\text{ kg}
\][/tex]
with a density of
[tex]\[
7000 \, \text{kg/m}^3.
\][/tex]
The pipe is rectangular with external dimensions
[tex]\[
12\text{ cm} \times 15\text{ cm}
\][/tex]
and internal (hollow) dimensions
[tex]\[
10\text{ cm} \times 12\text{ cm}.
\][/tex]
We will determine the length of the pipe by following these steps.
–––––––––––
Step 1. Calculate the Volume of Metal
The volume of metal available, [tex]$V$[/tex], can be computed by
[tex]\[
V = \frac{\text{mass}}{\text{density}}.
\][/tex]
Substituting the given values, we obtain
[tex]\[
V = \frac{63 \, \text{kg}}{7000 \, \text{kg/m}^3} = 0.009 \, \text{m}^3.
\][/tex]
–––––––––––
Step 2. Determine the Cross-Sectional Area of the Metal in the Pipe
The cross-sectional area of the pipe is the area of the external rectangle minus the area of the internal rectangle.
1. External Cross-Sectional Area:
The external dimensions are [tex]$12\text{ cm}$[/tex] by [tex]$15\text{ cm}$[/tex], so the external area, [tex]$A_{\text{ext}}$[/tex], is
[tex]\[
A_{\text{ext}} = 12 \times 15 = 180 \, \text{cm}^2.
\][/tex]
2. Internal (Hollow) Cross-Sectional Area:
The internal dimensions are [tex]$10\text{ cm}$[/tex] by [tex]$12\text{ cm}$[/tex], so the internal area, [tex]$A_{\text{int}}$[/tex], is
[tex]\[
A_{\text{int}} = 10 \times 12 = 120 \, \text{cm}^2.
\][/tex]
3. Metal Cross-Sectional Area:
Subtracting the internal area from the external area, we find
[tex]\[
A_{\text{metal}} = A_{\text{ext}} - A_{\text{int}} = 180 - 120 = 60 \, \text{cm}^2.
\][/tex]
Since we require areas in [tex]$\text{m}^2$[/tex] for consistency with the volume, note that
[tex]\[
1 \, \text{cm}^2 = 0.0001 \, \text{m}^2.
\][/tex]
Thus, converting the metal area to square meters, we have
[tex]\[
A_{\text{metal}} = 60 \times 0.0001 = 0.006 \, \text{m}^2.
\][/tex]
–––––––––––
Step 3. Calculate the Length of the Pipe
The volume of the metal used to make the pipe can be expressed as the product of the cross-sectional area of the metal and the length of the pipe, [tex]$L$[/tex]:
[tex]\[
V = A_{\text{metal}} \times L.
\][/tex]
We can solve for [tex]$L$[/tex]:
[tex]\[
L = \frac{V}{A_{\text{metal}}}.
\][/tex]
Substitute the values found in the previous steps:
[tex]\[
L = \frac{0.009 \, \text{m}^3}{0.006 \, \text{m}^2} \approx 1.5 \, \text{m}.
\][/tex]
–––––––––––
Final Answer:
The length of the pipe is approximately
[tex]\[
\boxed{1.5 \, \text{meters}}.
\][/tex]
[tex]\[
63\text{ kg}
\][/tex]
with a density of
[tex]\[
7000 \, \text{kg/m}^3.
\][/tex]
The pipe is rectangular with external dimensions
[tex]\[
12\text{ cm} \times 15\text{ cm}
\][/tex]
and internal (hollow) dimensions
[tex]\[
10\text{ cm} \times 12\text{ cm}.
\][/tex]
We will determine the length of the pipe by following these steps.
–––––––––––
Step 1. Calculate the Volume of Metal
The volume of metal available, [tex]$V$[/tex], can be computed by
[tex]\[
V = \frac{\text{mass}}{\text{density}}.
\][/tex]
Substituting the given values, we obtain
[tex]\[
V = \frac{63 \, \text{kg}}{7000 \, \text{kg/m}^3} = 0.009 \, \text{m}^3.
\][/tex]
–––––––––––
Step 2. Determine the Cross-Sectional Area of the Metal in the Pipe
The cross-sectional area of the pipe is the area of the external rectangle minus the area of the internal rectangle.
1. External Cross-Sectional Area:
The external dimensions are [tex]$12\text{ cm}$[/tex] by [tex]$15\text{ cm}$[/tex], so the external area, [tex]$A_{\text{ext}}$[/tex], is
[tex]\[
A_{\text{ext}} = 12 \times 15 = 180 \, \text{cm}^2.
\][/tex]
2. Internal (Hollow) Cross-Sectional Area:
The internal dimensions are [tex]$10\text{ cm}$[/tex] by [tex]$12\text{ cm}$[/tex], so the internal area, [tex]$A_{\text{int}}$[/tex], is
[tex]\[
A_{\text{int}} = 10 \times 12 = 120 \, \text{cm}^2.
\][/tex]
3. Metal Cross-Sectional Area:
Subtracting the internal area from the external area, we find
[tex]\[
A_{\text{metal}} = A_{\text{ext}} - A_{\text{int}} = 180 - 120 = 60 \, \text{cm}^2.
\][/tex]
Since we require areas in [tex]$\text{m}^2$[/tex] for consistency with the volume, note that
[tex]\[
1 \, \text{cm}^2 = 0.0001 \, \text{m}^2.
\][/tex]
Thus, converting the metal area to square meters, we have
[tex]\[
A_{\text{metal}} = 60 \times 0.0001 = 0.006 \, \text{m}^2.
\][/tex]
–––––––––––
Step 3. Calculate the Length of the Pipe
The volume of the metal used to make the pipe can be expressed as the product of the cross-sectional area of the metal and the length of the pipe, [tex]$L$[/tex]:
[tex]\[
V = A_{\text{metal}} \times L.
\][/tex]
We can solve for [tex]$L$[/tex]:
[tex]\[
L = \frac{V}{A_{\text{metal}}}.
\][/tex]
Substitute the values found in the previous steps:
[tex]\[
L = \frac{0.009 \, \text{m}^3}{0.006 \, \text{m}^2} \approx 1.5 \, \text{m}.
\][/tex]
–––––––––––
Final Answer:
The length of the pipe is approximately
[tex]\[
\boxed{1.5 \, \text{meters}}.
\][/tex]
