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.)
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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 \( \boxed{60} \).
