Final answer:
The total volume of the machine part is the sum of the volume of the cylinder, calculated as 108π cubic inches, and the volume of the half-sphere, which is 18π cubic inches, leading to a total of 126π cubic inches. However, this does not match the provided answer options.
Step-by-step explanation:
To find the total volume of a machine part consisting of a half-sphere and a cylinder, we need to calculate the volume of each part separately and then add them together.
First, we calculate the volume of the cylinder using the formula V = πr²h. The diameter of the cylinder is given as 6 inches, so the radius (r) is half of that, which is 3 inches. The height (h) of the cylinder is 12 inches.
The volume of the cylinder is:
Vcylinder = π × (3 in)2 × 12 in = π × 9 in2 × 12 in = 108π in3
Next, we calculate the volume of the half-sphere. The formula for the volume of a sphere is V = ⅔πr³, and since we have a half-sphere, we need to divide this by 2.
The volume of the half-sphere is:
Vhalf-sphere = ½ × ⅔π × (3 in)3 = ½ × ⅔π × 27 in3 = ½ × 36π in3 = 18π in3
Adding the volumes of the cylinder and half-sphere gives us the total volume:
Vtotal = π × 108 in3 + π × 18 in3 = 126π in3
However, none of the answer choices match 126π in3. There might be an error in the provided answer options or in the calculations.