Question: An angel investor evaluates a biotech startups 3D-printed cell culture scaffold shaped like a regular hexagonal prism with height $ h $ and base side length $ a $. If the base area scales by a factor of $ k $, what is the new volume in terms of $ h $, $ a $, and $ k $?

["An angel investor evaluates a biotech startups’ 3D-printed cell culture scaffold shaped like a regular hexagonal prism with height $ h $ and base side length $ a $. If the base area scales by a factor of $ k $, what is the new volume in terms of $ h $, $ a $, and $ k $?", "As tissue engineering and precision medicine push the boundaries of biotech innovation, novel scaffold designs are drawing attention from investors seeking next-generation solutions. One emerging structure gaining focus is the regular hexagonal prism—a 3D-printed architecture optimized for cell growth, valued for its strength and scalability. For investors assessing these startups, understanding how design changes like base area adjustments impact scalability and performance is essential. This article explores how scaling the base area by a factor of $ k $ affects volume, offering clear insights for informed decision-making in the U.S. biotech investment landscape.", "---", "Why This Design Matters in Current Biotech Trends", "The trend toward personalized tissue scaffolds is accelerating, driven by demand for better implants, drug testing platforms, and regenerative treatments. The hexagonal prism shape offers a compelling balance of structural integrity and surface area efficiency—key for supporting cell attachment and nutrient diffusion. As startups refine manufacturing processes to achieve lab-scale prototypes and push toward commercialization, investors scrutinize design scalability. One fundamental question arises: how do proportional changes in base area affect the overall volume, and what implications does this scaling have for performance and investment viability?", "---", "How Scaling Base Area Reshapes Volume", "The volume of a prism is calculated as base area multiplied by height. For this scaffold, the base is a regular hexagon with side length $ a $, so its area begins with the formula: \n$$ A = \frac{3\sqrt{3}}{2} a^2 $$ \nWhen the base area scales by a factor of $ k $, the new base area becomes $ k \cdot A = k \cdot \frac{3\sqrt{3}}{2} a^2 $. Since height $ h $ remains unchanged, the new volume $ V' $ is: \n$$ V' = (\ ext{new base area}) \ imes h = k \cdot \frac{"]