Final answer:
The calculations for the lateral surface area of a cone indicate that there might be an error in the question’s provided dimensions, as a typical paper cup would not have a diameter of 22 inches. The procedure explained uses the formula for the lateral surface area of a cone, but due to the possible typo in the question, none of the given answers match the result from the calculation.
Step-by-step explanation:
To calculate the lateral surface area of a cone, we can use the formula: lateral surface area = πrl, where r is the radius of the base and l is the slant height of the cone. The height of the cone is given as 3 inches, and the diameter is given as 22 inches, thus the radius r is half of the diameter, which is 11 inches.
To find the slant height l, we can use the Pythagorean theorem in the right triangle formed by the radius, height, and slant height of the cone: l = √(r² + h²). This gives us l = √(11² + 3²) = √(121 + 9) = √130 ≈ 11.4 inches.
Now we have all the values needed to calculate the lateral surface area: πrl = π(11 inches)(11.4 inches) ≈ 394.6 square inches. However, since the question seems to contain a typo regarding the dimensions of the cone (as a cup with a diameter of 22 inches is unlikely), we cannot provide a correct answer from the options provided. There might be a misunderstanding or an error in the question itself as the provided dimensions do not seem to be for a typical paper cup used for water.