A solid metal sphere of radius 4.2cm was melted and molten material used to make a cube. Find to 3 significant figures the length of the side of the cube.
∛(4/3 π) * 4.2 = 6.77
Working
To solve this problem, we will use the following formula:
Volume of a sphere = (4/3) * π * r^3
First, let's calculate the volume of the sphere using the given radius of 4.2 cm:
Volume of the sphere = (4/3) * 3.14159 * (4.2 cm)^3
Using a calculator:
Volume of the sphere ≈ 310.86 cm^3 (rounded to 3 significant figures)
Since the molten material was used to make a cube, the volume of the sphere must be equal to the volume of the cube:
Volume of the cube = side length^3
Therefore, we need to find the length of the side of the cube.
Setting the volumes equal:
310.86 cm^3 = side length^3
Now, we need to solve for the side length of the cube. Taking the cube root of both sides will give us the answer:
Cube root (310.86 cm^3) ≈ 6.75 cm (rounded to 3 significant figures)
Therefore, the length of the side of the cube is approximately 6.75 cm.
To find the length of the side of the cube, we can follow these steps:
Step 1: Find the volume of the sphere.
The volume of a sphere can be calculated using the formula:
V = 4/3 * π * r^3
where V is the volume and r is the radius of the sphere.
Plugging in the given radius of 4.2 cm:
V = 4/3 * π * (4.2 cm)^3
V ≈ 310.58 cm^3 (rounded to five decimal places)
Step 2: Calculate the volume of the cube.
Since the entire sphere was melted to form the cube, the volume remains the same.
Therefore, the volume of the cube is also approximately 310.58 cm^3.
Step 3: Find the length of the side of the cube.
The volume of a cube is calculated by taking the side length cubed:
V = s^3
where V is the volume and s is the side length of the cube.
To find the side length of the cube, we need to take the cube root of the volume:
s = ∛V
Plugging in the volume of 310.58 cm^3:
s = ∛310.58 cm^3 ≈ 6.30 cm
Therefore, to three significant figures, the length of the side of the cube is approximately 6.30 cm.