Final answer:
To find the total mass of the object, we divide the semicircle into two sections and calculate the mass of each section. Finally, we add the two masses together. The total mass of the object is approximately 25.12 g.
Step-by-step explanation:
To find the total mass of the object, we need to calculate the mass of each section and then add them together. Since the density changes linearly from 1 g/cm at the bottom to 3 g/cm at the top, we can divide the semicircle into two sections: the bottom half with density 1 g/cm and the top half with density 3 g/cm.
Step 1: Find the mass of the bottom half:
Mass = Density x Volume
Volume of semicircle = (1/2) x π x (radius^2)
Density of bottom half = 1 g/cm
Mass of bottom half = (1 g/cm) x [(1/2) x π x (4 cm^2)]
Step 2: Find the mass of the top half:
Volume of semicircle = (1/2) x π x (radius^2)
Density of top half = 3 g/cm
Mass of top half = (3 g/cm) x [(1/2) x π x (4 cm^2)]
Step 3: Add the masses of the bottom and top half together:
Total mass = Mass of bottom half + Mass of top half
Total mass = (1 g/cm x [(1/2) x π x (4 cm^2)]) + (3 g/cm x [(1/2) x π x (4 cm^2)])
Total mass = 2π cm^2
Total mass = 2π x 4 cm^2
Total mass = 8π cm^3
Using the approximation π = 3.14, we can calculate the total mass:
Total mass ≈ 8 x 3.14 g
Total mass ≈ 25.12 g
Therefore, the total mass of the object is approximately 25.12 g.