Let's break this problem down step by step.

a. First Offer's Post-Money Valuation

The post-money valuation is the value of the company after the venture capitalist's investment. It's calculated using the total number of shares in the company after the investment.

We know:

• The venture capitalist will invest $2.92 million.
• The venture capitalist will receive 1.12 million new shares.
• Currently, there are 10.07 million shares in the company.

Steps:

1. After the investment, the total number of shares will be: $\text{Total shares} = \text{Existing shares} + \text{New shares}$ $\text{Total shares} = 10.07 \, \text{million} + 1.12 \, \text{million} = 11.19 \, \text{million shares}$

2. The price per share, after the investment, is given by: $\text{Price per share} = \frac{\text{Investment amount}}{\text{New shares}}$ $\text{Price per share} = \frac{2.92 \, \text{million}}{1.12 \, \text{million}} = 2.6071 \, \text{dollars/share}$

3. The post-money valuation is the price per share multiplied by the total number of shares: $\text{Post-money valuation} = \text{Price per share} \times \text{Total shares}$ $\text{Post-money valuation} = 2.6071 \times 11.19 \, \text{million} = 29.1879 \, \text{million dollars}$

   Post-money valuation (rounded to nearest dollar):
   $\boxed{29,188,000} \, \text{dollars}$

Would you like me to move to part b or provide further clarification? Also, here are 5 related questions you could explore:

1. What is pre-money valuation, and how is it related to post-money valuation?
2. How does share dilution affect existing shareholders?
3. What are the risks of venture capital investment in startups?
4. How can a company negotiate better terms with venture capitalists?
5. How does issuing new shares affect a company’s stock price?

Tip: Always consider both the percentage dilution and the valuation terms when evaluating investment offers!