Question: A marine biologist is designing a conical fish shelter with a base radius of $ 2x $ units and a slant height of $ 5x $ units. What is the total lateral surface area of the cone in terms of $ \pi $ and $ x $?

["Question: A marine biologist is designing a conical fish shelter with a base radius of $ 2x $ units and a slant height of $ 5x $ units. What is the total lateral surface area of the cone in terms of $ \pi $ and $ x $?", "Answer:\nTo find the total lateral surface area of a cone, we use the formula:\n$$\n\ ext{Lateral Surface Area} = \pi r \ell\n$$\nwhere:\n- $ r $ is the base radius\n- $ \ell $ is the slant height", "From the question:\n- The base radius $ r = 2x $\n- The slant height $ \ell = 5x $", "Substituting these values into the formula:\n$$\n\ ext{Lateral Surface Area} = \pi (2x)(5x) = \pi \cdot 10x^2 = 10\pi x^2\n$$", "Thus, the total lateral surface area of the conical fish shelter is:\n$$\n\boxed{10\pi x^2}\n$$", "This precise calculation supports accurate design and material estimation for marine habitat structures, making it essential for marine biologists integrating engineering with ecology."]