Decide launch with CPA-profit trade-offs by segment

You receive an Excel with segment-level metrics for an A/B feature test. Decide whether to launch the feature for all users, only a segment, or not at all, and justify with calculations. Definitions: revenue = gross revenue; cost = non-marketing COGS; CPA = acquisition cost per new customer; profit = revenue − cost − (CPA × new_customers); LTV = lifetime gross margin dollars per customer; fixed costs unchanged; CPA applies only to new_customers; assume no other changes beyond what’s implied. Data (last 28 days): Segment A (e.g., new users) Variant | revenue | cost | CPA | new_customers | LTV Control | 120000 | 60000| 55 | 600 | 180 Feature | 100000 | 52000| 45 | 900 | 170 Segment B (e.g., existing users) Variant | revenue | cost | CPA | new_customers | LTV Control | 200000 |110000| 65 | 400 | 220 Feature | 230000 |125000| 60 | 450 | 230 Answer all parts: 1) Compute profit for Control vs Feature in each segment and the incremental profit (Feature − Control). 2) Compute LTV/CAC for each segment and variant (use CAC = CPA for new users; for existing users treat CPA as incremental marketing cost per incremental customer and explain your assumption). 3) If only one global decision is allowed (ship to 100% or to 0%), what CPA cap for Segment A would make a full launch profit-neutral vs Control? Show the formula and the numeric threshold. 4) If Segment B represents x% of total traffic and Segment A is (100 − x)%, derive the minimum x at which a full launch (both segments) is profit-improving overall. Provide x as a percentage and your algebra. 5) Recommend: launch to B only, launch to all, or hold. Include 2 key risks/trade-offs (e.g., scale limits, cannibalization) and 1 follow-up analysis you would run before a final go/no-go.