Question

Шар,объем которого равен 15п, вписан в куб. Найдите объем куба

Answer 1

V(шара)=4πR³/3
15π=4πR³/3
45π=4πR³  ⇒    45=4R³    R=∛45/4
a=2R
V=a³=(2R)³=8R³=8·(45/4)=2·45=90 куб. ед

Answer 2

Task: Finding the volume of a cube with a sphere inscribed in it.

To find the volume of a cube, we need to determine the length of its side. In this case, we are given that the volume of the inscribed sphere is 15π.

Let's break down the problem into smaller steps:

1. Find the radius of the inscribed sphere. 2. Determine the length of the side of the cube using the radius. 3. Calculate the volume of the cube using the side length.

Step 1: Finding the radius of the inscribed sphere The volume of a sphere is given by the formula: V = (4/3)πr³, where V is the volume and r is the radius.

In this case, we are given that the volume of the sphere is 15π. So, we can set up the equation:

15π = (4/3)πr³

To solve for r, we can cancel out the π and simplify the equation:

15 = (4/3)r³

Now, we can solve for r by taking the cube root of both sides:

r = ∛(15 * (3/4))

r ≈ ∛(11.25) r ≈ 2.24

So, the radius of the inscribed sphere is approximately 2.24.

Step 2: Determining the length of the side of the cube In a cube, the diameter of the inscribed sphere is equal to the length of the side of the cube. Since the radius of the sphere is 2.24, the diameter is 2 * 2.24 = 4.48.

Therefore, the length of the side of the cube is also 4.48.

Step 3: Calculating the volume of the cube The volume of a cube is given by the formula: V = s³, where V is the volume and s is the length of the side.

In this case, the length of the side is 4.48, so we can calculate the volume of the cube:

V = 4.48³ V ≈ 86.41

Therefore, the volume of the cube is approximately 86.41 cubic units.

So, the answer is: The volume of the cube is approximately 86.41 cubic units.