A souvenir consisting of a cube with an inserted sphere is made to represent Earth. Both pieces are made out of solid crystal. The sphere replaces one quarter of the volume of the cube. Calculate the volume of the souvenir piece.
The sphere has diameter of 4 cm and the cube had side length of 4cm.
I know to use formulas for volume for both shapes but after that I am completely lost. Please Help!!!
Thank you.
81.5
cube has volume 4^3 = 64
sphere has volume π/6 * 4^3 = 64π/6
Now just subtract the volumes:
64(1 - π/6)
Actually, the wording is a bit vague. Since the sphere has replaced some of the cube, the space is still occupied, so the whole cube constitutes the souvenir. volume=64 cm^3
To calculate the volume of the souvenir piece, you'll need to find the volume of both the cube and the sphere and then subtract one quarter of the sphere's volume from the cube's volume. Here's how you can do it step by step:
1. Calculate the volume of the cube:
The formula to calculate the volume of a cube is V = s^3, where s is the length of a side of the cube.
In this case, the side length of the cube is 4 cm. So, the volume of the cube is V_c = 4^3 = 64 cm^3.
2. Calculate the volume of the sphere:
The formula to calculate the volume of a sphere is V = (4/3)πr^3, where r is the radius of the sphere.
The diameter of the sphere is given as 4 cm, so the radius is half of the diameter, which is 4/2 = 2 cm.
Substituting the values in the formula, we get V_s = (4/3)π(2)^3 = (4/3)(π)(8) = 32(π/3) cm^3.
3. Calculate one quarter of the sphere's volume:
To find one quarter of the sphere's volume, divide the volume of the sphere by 4.
So, one quarter of the sphere's volume is (1/4)(32(π/3)) = 8(π/3) cm^3.
4. Calculate the volume of the souvenir piece:
Finally, subtract the volume of one quarter of the sphere from the volume of the cube.
The volume of the souvenir piece is V_souvenir = V_c - 8(π/3) = 64 cm^3 - 8(π/3) cm^3.
This should give you the volume of the souvenir piece.