A hollow cylindrical iron pipe with external and internal radii 8cm and 6cm respectively and length is 35cm is melted and recast into a solid wire of thickness 2.8cm. Find the length of wire
volume of original pipe = π(8^2 - 6^2)(35)
"solid wire of thickness 2.8cm" to me suggests that the diameter is 2.8
so the radius would be 1.4
length of new wire = π(1.4)^2 (L) , where L is the length of the new wire
π(1.4)^2 (L) = π(8^2 - 6^2)(35)
L = (8^2 - 6^2)(35) / 1.96
= 500
looking at the robot's last two lines ....
...
= (504 x 35) / (7.84)
= 6,420 cm
How can it even get that arithmetically ???
To find the length of the wire, we need to calculate the volume of the cylindrical pipe and then divide it by the cross-sectional area of the wire.
The volume of the cylindrical pipe can be calculated as the difference between the volumes of the outer and inner cylinders.
The volume of the outer cylinder can be found using the formula: V_outer = π * r_outer^2 * h, where r_outer is the external radius and h is the length of the pipe.
V_outer = π * (8cm)^2 * 35cm
The volume of the inner cylinder can be found using the same formula with the inner radius:
V_inner = π * r_inner^2 * h
V_inner = π * (6cm)^2 * 35cm
Now, let's calculate the volumes:
V_outer = π * (8cm)^2 * 35cm
= 2240π cm^3
V_inner = π * (6cm)^2 * 35cm
= 1260π cm^3
The volume of the wire can be calculated as the difference between the volumes of the outer and inner cylinders:
V_wire = V_outer - V_inner
= 2240π cm^3 - 1260π cm^3
= 980π cm^3
The thickness of the wire is given as 2.8 cm, which is equivalent to the difference between the outer and inner radii:
Thickness = r_outer - r_inner
2.8cm = 8cm - 6cm
Therefore, the radius of the wire, r_wire, is 2.8/2 = 1.4 cm.
Now, let's find the cross-sectional area of the wire:
Area_wire = π * r_wire^2
= π * (1.4cm)^2
= 1.4^2π cm^2
Finally, we can calculate the length of the wire by dividing the volume by the cross-sectional area:
Length_wire = V_wire / Area_wire
= (980π cm^3) / (1.4^2π cm^2)
Simplifying, we get:
Length_wire = (980 cm^3) / (1.96 cm^2)
= 500 cm
Therefore, the length of the wire is 500 cm.
To find the length of the wire, we need to consider the volume of both the cylindrical pipe and the solid wire. Since the volume remains constant during the transformation, we can equate the volume of the pipe to the volume of the wire.
Volume of the pipe:
The volume of the pipe can be calculated using the formula for the volume of a cylinder: V_pipe = π * (R^2 - r^2) * h,
where R is the external radius, r is the internal radius, and h is the height or length of the pipe.
V_pipe = π * (8^2 - 6^2) * 35
V_pipe = π * (64 - 36) * 35
V_pipe = π * (28) * 35
V_pipe = 980π cm^3
Volume of the wire:
The volume of the wire can be calculated using the formula for the volume of a cylinder: V_wire = π * R^2 * h_wire,
where R is the radius of the wire and h_wire is the length of the wire.
Given that the thickness of the wire is 2.8 cm, we can calculate the radius of the wire: R = r + thickness = 6 + 2.8 = 8.8 cm.
V_wire = π * (8.8^2) * h_wire
V_wire = 77.44π * h_wire cm^3
Since the volume remains constant, we can equate the two volume formulas:
980π = 77.44π * h_wire
Dividing both sides of the equation by 77.44π:
12.65 = h_wire
Therefore, the length of the wire is approximately 12.65 cm.
Length of wire = (π x (8^2 - 6^2) x 35) / (π x 2.8^2)
= (504 x 35) / (7.84)
= 6,420 cm