nment A company charges \( \$ 20 \) to make one monogrammed shirt, but reduces this cost by \( \$ 0.10 \) per shirt for each additional shirt ordered up to 100 shirts. If the of an order is \( \$ 846 \), how many shirts were ordered? If the cost of an order is \( \$ 846.2 .145 \), \( \square \) shirts were ordered. (Simplify your answer.)

To find out how many shirts were ordered, we can start with the total cost equation. The cost of each shirt decreases by $0.10 for each additional shirt ordered. So, if \( n \) is the number of shirts ordered, the cost per shirt becomes \[ 20 - 0.10(n - 1) \] Thus, the total cost \( C \) for \( n \) shirts is \[ C = n \cdot (20 - 0.10(n - 1)) = n(20 - 0.10n + 0.10) = n(20.10 - 0.10n). \] Setting this equal to $846 gives: \[ n(20.10 - 0.10n) = 846. \] Expanding and rearranging gives the quadratic equation: \[ 0.10n^2 - 20.10n + 846 = 0. \] Multiplying through by 10 to eliminate decimals yields: \[ n^2 - 201n + 8460 = 0. \] Applying the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ n = \frac{201 \pm \sqrt{(-201)^2 - 4 \cdot 1 \cdot 8460}}{2 \cdot 1}. \] Calculating inside the square root: \[ n = \frac{201 \pm \sqrt{40401 - 33840}}{2}, \] \[ n = \frac{201 \pm \sqrt{6561}}{2}, \] \[ n = \frac{201 \pm 81}{2}. \] This results in two possible solutions: 1. \( n = \frac{282}{2} = 141 \) (not possible as it exceeds 100). 2. \( n = \frac{120}{2} = 60 \). So, the number of shirts ordered is \( \boxed{60} \).