A sphere with a radius of 5 inches is glued inside a cube with a side length of 10 inches. What is the probability that a point chosen at random inside the cube is also inside the sphere? Round the probability to the nearest hundredth of a percent.Probability: about %
Question
A sphere with a radius of 5 inches is glued inside a cube with a side length of 10 inches. What is the probability that a point chosen at random inside the cube is also inside the sphere? Round the probability to the nearest hundredth of a percent.Probability: about %
Solution
To solve this problem, we need to find the ratio of the volume of the sphere to the volume of the cube.
Step 1: Calculate the volume of the sphere. The formula for the volume of a sphere is (4/3)πr³. Given that the radius of the sphere is 5 inches, the volume of the sphere is (4/3)π(5)³ = 500π/3 cubic inches.
Step 2: Calculate the volume of the cube. The formula for the volume of a cube is s³. Given that the side length of the cube is 10 inches, the volume of the cube is 10³ = 1000 cubic inches.
Step 3: Calculate the ratio of the volume of the sphere to the volume of the cube. The ratio is (500π/3) / 1000 = 500π/3000 = π/6.
Step 4: Convert the ratio to a probability. The probability is the ratio expressed as a decimal. Therefore, the probability is π/6 ≈ 0.5236.
Step 5: Convert the decimal to a percentage and round to the nearest hundredth of a percent. The percentage is 0.5236 * 100% = 52.36%. Rounded to the nearest hundredth of a percent, the probability is approximately 52.36%.
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