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Question: An industrial designer is creating a conical water cup with a base radius of $ 3x $ units and height $ 4x $ units. What is the total surface area of the cup, including the base?
Solution: The total surface area of a cone includes the lateral (curved) surface area and the base area. The formula for total surface area is:
where $ r $ is the base radius and $ l $ is the slant height. Given $ r = 3x $, and height $ h = 4x $, we first find the slant height $ l $ using the Pythagorean theorem:

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