A company produces products that follow a conic section pattern defined by the polar equation \[ r = \frac{4}{3 + 2 \cos \theta} \]. Each product costs $5 to produce and is sold for $12. If the company produces and sells 100 units, what is the total profit made? Additionally, identify the conic section represented by the given polar equation. Let's think step by step and output the final answer within \boxed{}.
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