A cosmetics company is developing a new product that combines three active ingredients: Ingredient X, Ingredient Y, and Ingredient Z. The table below shows the cost per gram and the density for each ingredient.
| Ingredient | Cost per gram ($) | Density (g/cm³) |
|---|---|---|
| X | 0.50 | 1.2 |
| Y | 0.80 | 0.9 |
| Z | 0.30 | 1.5 |
The total cost, , in dollars, for a batch of the new product is given by the equation , where , , and are the volumes, in cm³, of Ingredient X, Ingredient Y, and Ingredient Z, respectively. If a batch contains 100 cm³ of Ingredient X and 50 cm³ of Ingredient Y, and the total cost of the batch is $145.50, what is the cost, in dollars, of the Ingredient Z used in this batch?
