The Psychiatric Hospitals P&L is a classic two-sided marketplace that matches Healthcare Facilities (HCFs) who need last minute shifts filled with Healthcare Professionals (HCPs) nearby who happen to have availability. You know that this marketplace is heavily supply-constrained (i. e. we don't have enough active nurses and open shifts are going unfilled). You know that paid marketing is the primary acquisition channel, so you ask marketing to get more nurses on the platform. Marketing asks you the series of questions below to guide their work. To answer their questions, here's the data you have:

Our HCP acquisition funnel has four parts: prospective HCPs fill out a job application, then go through onboarding (i. e. submitting proof of licensure & other documentation), and then go on to work shifts on the platform. Marketing data, on average, over the last 6 months for the HCP acquisition side:

Marketing spend per completed application: $20
Percent of completed applications that are Valid: 80%
Conversion rate from Valid Application to Fully Onboarded: 25%
Conversion rate from Fully Onboarded to "Worked one or more shifts": 33%
Revenue & Churn

Average net revenue to Clipboard Health per shift worked: $50
Average monthly headcount churn: 11%
First month HCP user behavior:

Healthcare professionals who work one shift usually end up working the following amount of shifts in their first 30 days:
33% worked a total of 3 shifts
33% worked a total of 8 shifts
20% worked a total of 12 shifts
14% worked a total of 16 shifts

Required:
a. What is the lifetime value of each nurse who works one or more shifts on the platform?
b. Assuming you're trying to hit a LTV/CAC ratio of 3, how much should marketing be willing to spend up to (on average) per new completed application?
c. For the last 6 months, marketing has been holding steady attracting roughly 486 new applicants a month, and as of today we happen to have 300 working nurses on the platform generating roughly $200k in monthly revenue. You want to go faster, and have 900 or more working nurses on the platform by the board meeting in 90 days, obviously while still adhering to the board's LTV/CAC ratio goal of 3 along the way. Let's assume, as is often the case, that larger cohorts perform worse. For our purposes, let's assume that each time we increase our monthly cohort size by 2x the cohort performs 10% worse in terms of first-month revenue once converted (we can assume that conversion rate stays the same).