A 2 inch by 2 inch by 2 inch block of wood has 1 inch diameter hole drilled. Through the center. Find the total volume and surface area. Round to the nearest tenth.
the original block has surface 6*2^2
subtract the two circles removed: 2*πr^2
add the curved surface of the hole: 2πrh
where r = 1/2 and h=2
the volume is of course 2^3 - πr^2h
To find the total volume and surface area of the block of wood with a drilled hole, we will need to break it down into different components: the solid cube and the cylindrical hole.
First, let's calculate the volume of the cube:
The volume of a cube is given by V_cube = length x width x height. Since the block of wood is a cube, with a length, width, and height of 2 inches each, we can calculate the volume as V_cube = 2 inches x 2 inches x 2 inches = 8 cubic inches.
Next, let's calculate the volume of the cylindrical hole:
The volume of a cylinder is given by V_cylinder = π x radius^2 x height. In this case, the radius is given as 1 inch (since the hole diameter is 1 inch, the radius would be half of that) and the height is the total length of the cube, which is 2 inches. Plugging these values into the formula gives V_cylinder = π x (1 inch)^2 x 2 inches = 2π cubic inches.
Now, to find the total volume, we need to subtract the volume of the cylindrical hole from the volume of the cube:
Total volume = V_cube - V_cylinder = 8 cubic inches - 2π cubic inches.
We can leave the answer in terms of π or use an approximation of 3.14 for π and calculate the value numerically.
To find the surface area, we need to consider the surfaces of the cube and the cylindrical hole separately.
Surface area of the cube:
The surface area of a cube is given by SA_cube = 6 x (length x width) = 6 x (2 inches x 2 inches) = 24 square inches.
Surface area of the cylindrical hole:
The surface area of a cylinder includes the curved part (lateral surface area), which is given by LA_cylinder = 2π x radius x height, and the top and bottom circular faces (base area), which is given by BA_cylinder = 2 x π x radius^2. In this case, the height is 2 inches, and the radius is 1 inch. Plugging these values into the formulas gives LA_cylinder = 2π x 1 inch x 2 inches = 4π square inches, and BA_cylinder = 2 x π x (1 inch)^2 = 2π square inches.
The total surface area is obtained by adding together the surface area of the cube and the surface area of the cylindrical hole:
Total surface area = SA_cube + LA_cylinder + BA_cylinder = 24 square inches + 4π square inches + 2π square inches.
Again, we can leave the answer in terms of π or approximate π as 3.14 and calculate the value numerically.