Pregunta
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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.)

Ask by Ortiz Davison. in the United States
Mar 14,2025

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60 shirts were ordered.

Solución

We let \( x \) be the number of shirts ordered. The company’s pricing rule is that the cost to make one shirt is reduced by \(\$0.10\) for every additional shirt ordered. In other words, the cost per shirt when \( x \) shirts are ordered becomes \[ 20 - 0.10(x-1). \] Simplify the expression: \[ 20 - 0.10(x-1)=20-0.10x+0.10=20.10-0.10x. \] The total cost of the order, \( C \), is the number of shirts times the cost per shirt: \[ C = x(20.10-0.10x). \] For the first part of the problem the total cost is given as \(\$846\). Thus the equation is: \[ x(20.10-0.10x)=846. \] To clear decimals, multiply both sides by 10: \[ x(201- x)=8460. \] Expanding, we obtain: \[ 201x-x^2=8460. \] Writing in standard quadratic form: \[ x^2-201x+8460=0. \] Next, we compute the discriminant: \[ \Delta = 201^2-4\cdot 1\cdot8460. \] Calculate: \[ 201^2=40401, \] \[ 4\cdot8460=33840, \] so \[ \Delta=40401-33840=6561. \] Since \[ \sqrt{6561}=81, \] the solutions for \( x \) are given by: \[ x=\frac{201\pm 81}{2}. \] Thus, \[ x=\frac{201+81}{2}=\frac{282}{2}=141 \quad \text{or} \quad x=\frac{201-81}{2}=\frac{120}{2}=60. \] Because the discount applies only for orders up to 100 shirts, the value \( x=141 \) is not allowable. Therefore, the number of shirts ordered is \[ \boxed{60}. \]

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The Deep Dive

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 \( \boxed{60} \).

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