The density of carbon in the form of diamonds is 3.51 g/cm^3. If you have a 40- carat spherical diamond, what is the diameter of the diamond in millimeters? (1 carat=0.200 g)
mass diamond = 40 carats x (0.200g/carat) = ?
mass = volum x density. You know density and mass, solve for volume.
Volumer of sphere = (4/3)*pi*r^3
Solve for r, then diameter = 2r and the answer will be in cm.
Convert cm to mm.
To find the diameter of the diamond, we can use the formula for the volume of a sphere:
V = (4/3) * π * r^3
First, let's convert the mass of the diamond from carats to grams:
Mass of the diamond = 40 carats * 0.200 g/carat
Mass of the diamond = 8 g
Next, we can find the radius of the diamond using the density formula:
Density = Mass / Volume
Rearranging the formula, we get:
Volume = Mass / Density
Volume = 8 g / 3.51 g/cm^3
Volume = 2.277166 g/cm^3
Now, let's substitute the volume into the sphere formula to find the radius:
V = (4/3) * π * r^3
2.277166 g/cm^3 = (4/3) * π * r^3
Solving for r^3, we get:
r^3 = (2.277166 g/cm^3 * 3) / (4π)
r^3 = 1.7078745 g/cm^3 / π
r^3 = 0.54315608 g/cm^3
Taking the cubic root of both sides, we find:
r ≈ 0.8216 cm
Finally, we can find the diameter of the diamond by multiplying the radius by 2:
Diameter = 2 * r
Diameter ≈ 2 * 0.8216 cm
Diameter ≈ 1.6432 cm
To convert the diameter to millimeters, we'll multiply by 10:
Diameter ≈ 1.6432 cm * 10 mm/cm
Diameter ≈ 16.432 mm
Therefore, the diameter of the diamond is approximately 16.432 millimeters.
To find the diameter of the diamond, we can use the formula for the volume of a sphere. First, we need to convert the mass of the diamond from carats to grams.
Given:
Density of carbon in the form of diamonds = 3.51 g/cm³
Mass of the diamond = 40 carats
1 carat = 0.200 g
Step 1: Convert the mass of the diamond to grams.
Mass of the diamond = 40 carats × 0.200 g/carat = 8 g
Step 2: Calculate the volume of the diamond.
The formula for the volume of a sphere is V = (4/3)πr³, where r is the radius of the sphere.
We know that density = mass/volume, so we can rearrange this equation to solve for volume: volume = mass/density.
Volume of the diamond = 8 g / 3.51 g/cm³ = 2.279 cm³
Step 3: Calculate the radius of the diamond.
The volume of a sphere is given by the formula V = (4/3)πr³. Rearranging this formula, we get r = (3V/4π)^(1/3).
Radius of the diamond = (3 × 2.279 cm³ / (4 × 3.14))^(1/3) = 0.924 cm
Step 4: Calculate the diameter of the diamond.
The diameter of a sphere is twice the radius.
Diameter of the diamond = 2 × 0.924 cm = 1.848 cm
Step 5: Convert the diameter to millimeters.
1 cm = 10 mm, so we can convert the diameter from centimeters to millimeters by multiplying by 10.
Diameter of the diamond = 1.848 cm × 10 mm/cm = 18.48 mm
Therefore, the diameter of the spherical diamond is approximately 18.48 mm.