I really need help on this question.
A goldsmith makes a wedding band by taking a gold ball and drilling a wide hole through it. The wedding band has to be 1 cm wide and the hole has to have a diameter of 2 cm.
Given that gold weighs 19.32g/cm^3, how heavy will the wedding band be?
There is actually a diagram to come with it, but i can't paste web addressess here.
the equation i came up with is
V = 2pi integral{0 to 1/2} (1 - x)dx
and then mass = (volume)(19.32g/cm^3)
does this seem right?
http://www.sfu.ca/~adebened/funstuff/sphere_cyl.pdf#search=%22sphere%20hole%20drilled%20volume%22
does 2.529 g seem right to my problem?
No, the formula to use is
V=4/3 * pi * h3 where h = .5
V = approx. 0.52359 cm3, so multiply that by the density.
mass = 0.52359 cm3 * 19.32g/cm33 = approx 10.12gm
Your equation and approach are incorrect. To find the volume of the wedding band, you need to subtract the volume of the hole from the volume of the gold ball.
The volume of the gold ball can be calculated using the formula for the volume of a sphere:
V_sphere = (4/3) * pi * r^3
Given that the diameter of the gold ball is 2 cm, the radius (r) is 1 cm. Substituting the values into the formula, we get:
V_sphere = (4/3) * pi * (1 cm)^3
= (4/3) * pi * 1 cm^3
= (4/3) * pi cm^3
Now, let's calculate the volume of the hole. The hole has a diameter of 2 cm, so the radius (r_hole) is 1 cm. The volume of the hole can be calculated as the difference between the volume of the sphere and the volume of the hole:
V_hole = V_sphere - V_cylinder
To find V_cylinder, we use the formula for the volume of a cylinder:
V_cylinder = pi * r_hole^2 * h
Given that the hole has a diameter of 2 cm, the radius (r_hole) is 1 cm. We also know that the width of the wedding band is 1 cm. So, substituting the values into the formula, we get:
V_cylinder = pi * (1 cm)^2 * 1 cm
= pi cm^3
Now, substituting the values of V_cylinder and V_sphere into the formula for V_hole:
V_hole = V_sphere - V_cylinder
= (4/3) * pi cm^3 - pi cm^3
= (4/3 - 1) * pi cm^3
= (1/3) * pi cm^3
Finally, to find the mass of the wedding band, we multiply the volume of the hole by the density of gold:
mass = V_hole * density
Given that the density of gold is 19.32 g/cm^3, we can substitute the values:
mass = (1/3) * pi cm^3 * 19.32 g/cm^3
= (1/3) * pi * 19.32 g.
Calculating this expression gives us the correct answer for the weight of the wedding band.