Four identical metal spheres, each with radius 6 cm, are melted and reshaped into one big sphere. Fine the radius of the new sphere.
Find out the volume of each sphere.
V = (4/3)(pi)(r^3)
Multiply by 4.
Use the same formula to find the radius of the new sphere.
To find the radius of the new sphere, we can use the concept of volume conservation. The volume of each sphere is given by the formula:
V = (4/3) * π * r^3
where r is the radius of each sphere.
Since all four spheres are identical, the total volume of all four spheres is equal to the volume of the new sphere:
4V = (4/3) * π * R^3
where R is the radius of the new sphere.
To determine the radius of the new sphere, we need to solve the equation for R.
Step 1: Calculate the volume of each of the four spheres:
V = (4/3) * π * (6 cm)^3
V = (4/3) * π * (216 cm^3)
V = 288π cm^3
Step 2: Substitute the volume of each sphere into the equation and solve for R:
4 * (288π cm^3) = (4/3) * π * R^3
1152π cm^3 = (4/3) * π * R^3
Cancel out the π term on both sides:
1152 cm^3 = (4/3) * R^3
Multiply both sides by 3/4 to isolate R^3:
(3/4) * 1152 cm^3 = R^3
864 cm^3 = R^3
Take the cube root of both sides to find R:
R ≈ (cube root of 864) cm
Using a calculator, the approximate value of the cube root of 864 is 9.267.
Therefore, the radius of the new sphere is approximately 9.267 cm.
To find the radius of the new sphere, we can use the concept of volume. The volume of a sphere is given by the formula:
V = (4/3)πr^3
Since we have four identical spheres, each with a radius of 6 cm, we can find the combined volume of the four spheres:
V_total = 4 * V_individual = 4 * (4/3)π(6^3)
V_total = (4/3)π * 4 * 6^3
Simplifying this expression:
V_total = (4/3)π * 4 * 216
V_total = (4/3)π * 864
Now, we can find the radius of the new sphere using the formula for volume:
V = (4/3)πr^3
Plugging in the given value for the total volume (V_total) and solving for the radius (r_new):
(4/3)πr_new^3 = (4/3)π * 864
r_new^3 = 864
Taking the cube root of both sides to solve for r_new:
r_new = ∛864
Using a calculator to simplify this expression, we find:
r_new ≈ 9.08 cm
Therefore, the radius of the new sphere is approximately 9.08 cm.