An unsharpened pencil is a regular hexagonal prism of wood with a cylinder of graphite in the middle. The hexagon has a side of 4mm. The graphite is 2mm in diameter. The pencil is 180 mm long. Determine the volume of wood in the pencil.
To determine the volume of wood in the pencil, we need to calculate the volume of the hexagonal prism and subtract the volume of the graphite cylinder.
1. Calculate the volume of the hexagonal prism:
The hexagonal prism can be divided into six congruent triangles, each with one side length of 4mm and a height of 180mm. To find the area of one triangular face, we can use the formula for the area of an equilateral triangle:
Area = (sqrt(3) / 4) * s^2,
where s is the length of one side of the equilateral triangle.
Substituting s = 4mm into the formula, we get:
Area = (sqrt(3) / 4) * (4mm)^2 = (sqrt(3) / 4) * 16mm^2 = 4 * sqrt(3) mm^2.
The total area of all six triangular faces is 6 times this value, so:
Total area of hexagonal prism = 6 * 4 * sqrt(3) mm^2.
To find the volume of the hexagonal prism, we need to multiply the area by the height:
Volume of hexagonal prism = Total area of hexagonal prism * height
= (6 * 4 * sqrt(3) mm^2) * 180mm
= 432 * sqrt(3) mm^3.
2. Calculate the volume of the graphite cylinder:
The volume of a cylinder can be calculated using the formula:
Volume = pi * r^2 * h,
where r is the radius of the base (which is half the diameter) and h is the height.
The radius of the graphite cylinder is 2mm / 2 = 1mm.
Substituting these values into the formula, we get:
Volume of graphite cylinder = pi * (1mm)^2 * 180mm
= 180 * pi mm^3.
3. Calculate the volume of wood in the pencil:
To find the volume of wood in the pencil, we subtract the volume of the graphite cylinder from the volume of the hexagonal prism:
Volume of wood = Volume of hexagonal prism - Volume of graphite cylinder
= (432 * sqrt(3) mm^3) - (180 * pi mm^3).
Now, you can calculate this expression to find the volume of wood in the pencil.