a cylinder of maximum volume is cut off from a solid cuboid with square box of side 14 cm and height 20 cm find the volume of metal
To find the volume of the metal, we need to calculate the volume of the cuboid and subtract the volume of the cylinder that was cut off.
First, let's find the volume of the cuboid:
Volume of a cuboid = length × width × height
In this case, the length, width, and height of the cuboid are all given as 14 cm, so we can substitute those values into the formula:
Volume of cuboid = 14 cm × 14 cm × 20 cm
Now, we need to find the volume of the cylinder that was cut off from the cuboid. To do this, we need to determine the dimensions of the cylinder.
Since the cylinder has a maximum volume, it means that the diameter of the cylinder should be equal to the side length of the square base of the cuboid, which is 14 cm. The height of the cylinder should be equal to the height of the cuboid, which is 20 cm.
The volume of a cylinder is given by the formula:
Volume of cylinder = π × r^2 × h
Substituting in the values:
Volume of cylinder = π × (14 cm/2)^2 × 20 cm
To find the volume of the metal, we subtract the volume of the cylinder from the volume of the cuboid:
Volume of metal = Volume of cuboid - Volume of cylinder
Now, let's calculate:
Volume of metal = (14 cm × 14 cm × 20 cm) - (π × (14 cm/2)^2 × 20 cm)
Simplifying the equation:
Volume of metal = 3920 cm^3 - (3.14 × 7 cm × 7 cm × 20 cm)
Calculating further:
Volume of metal = 3920 cm^3 - 8624 cm^3
Finally, you can subtract to find the volume of metal:
Volume of metal = -4704 cm^3
Note: The volume of the metal is negative because the volume of the cylinder cut off from the cuboid is greater than the volume of the cuboid itself. It means that the metal taken out is more than the metal present in the original cuboid.
well, it appears that the cylinder has diameter 14 and height 20.
v = pi r^2 h, right?