Question at position 20 A cylinder has a height of 4 meters and a radius of 1.5 meters. What is the approximate radius of a sphere that has the same surface area as thecylinder? AnswerA cylinder has a height of 4 meters and a radius of 1.5 meters. What is the approximate radius of a sphere that has the same surface area as thecylinder? [input]题目解析

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BlackTom AI · Accepted Answer

First, identify the given cylinder dimensions: radius r = 1.5 m and height h = 4 m. The surface area (SA) of a right cylinder is given by SA = 2πr(h + r). Substituting in the values, SA = 2π(1.5)(4 + 1.5) = 3π(5.5) = 16.5π. Numerically, 16.5π ≈ 51.8 square meters.

Next, consider the sphere that has the same surface area. The surface area of a sphere is SA = 4πR^2, where R is the sphere’s radius. Set 4πR^2 equal to the cylinder’s surface area: 4πR^2 = 16.5π. Dividing both sides by 4π gives R^2 = 16.5π / (4π) = 16.5 / 4 = 4.125.

Taking the square root yields R = √4.125 ≈ 2.032... meters. Rounding to two decimal places gives R ≈ 2.04 meters.

To check for consistency: the cylinder’s surface area calculation used the full SA formula, not just lateral area, so the comparison is apples-to-apples with the sphere’s total surface area. Missteps to avoid include treating the sphere’s area as 4πR or using incorrect formulas like SA of a cylinder that omits the top and bottom, which would lead to a smaller SA and a different R. Also ensure the rounding reflects the same precision as the given answer (two decimals here).

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