A cube has an edge length of 68 inches. A solidsphere with a radius of 34 inches is inside the cube,such that the sphere touches the center of each faceof the cube. To the nearest cubic inch, what is thevolume of the space in the cube not taken up by thesphere?A) 149,796B) 164,500C) 190,955D) 310,800
Question
A cube has an edge length of 68 inches. A solidsphere with a radius of 34 inches is inside the cube,such that the sphere touches the center of each faceof the cube. To the nearest cubic inch, what is thevolume of the space in the cube not taken up by thesphere?A) 149,796B) 164,500C) 190,955D) 310,800
Solution
To solve this problem, we first need to find the volume of the cube and the volume of the sphere, and then subtract the volume of the sphere from the volume of the cube.
Step 1: Find the volume of the cube The formula for the volume of a cube is side length cubed. So, the volume of the cube is 68^3 = 314,432 cubic inches.
Step 2: Find the volume of the sphere The formula for the volume of a sphere is 4/3 * π * radius^3. So, the volume of the sphere is 4/3 * π * 34^3 = 164,636 cubic inches (rounded to the nearest cubic inch).
Step 3: Subtract the volume of the sphere from the volume of the cube 314,432 - 164,636 = 149,796 cubic inches.
So, the volume of the space in the cube not taken up by the sphere is approximately 149,796 cubic inches. Therefore, the answer is (A) 149,796.
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