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(Linear cost model) Christian Jiménez determines that if he produces 100 items the total cost is $500, while if he produces 150 items the total cost is $600. Assuming the production-cost model is linear, determine the fixed cost and variable costs. What will be the cost of producing 200 items?

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Answer to a math question (Linear cost model) Christian Jiménez determines that if he produces 100 items the total cost is $500, while if he produces 150 items the total cost is $600. Assuming the production-cost model is linear, determine the fixed cost and variable costs. What will be the cost of producing 200 items?

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Murray
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Para determinar los costos fijos y variables en un modelo de costos lineal, podemos usar la información proporcionada para crear un sistema de ecuaciones. Denotemos el costo fijo como \( F \) y el costo variable por artículo como \( V \). De los datos dados: 1. Cuando se producen 100 artículos, el costo total es de $500. 2. Cuando se producen 150 artículos, el costo total es de $600. Podemos plantear las siguientes ecuaciones: 1. \(100V + F = 500\) 2.\(150V + F = 600\) Ahora, resolvamos este sistema de ecuaciones para encontrar \( V \) y \( F \). Restar la primera ecuación de la segunda nos da: \( 150V + F - (100V + F) = 600 - 500 \) \(50V = 100\) \( V = 2 \) Ahora que tenemos \( V \), podemos sustituirlo nuevamente en la primera ecuación para encontrar \( F \): \( 100(2) + F = 500 \) \( 200 + F = 500 \) \( F = 300 \) Entonces, el costo fijo \( F \) es $300 y el costo variable \( V \) es $2 por artículo. Para encontrar el costo de producir 200 artículos, usamos la ecuación: \( \text{Costo total} = V \times \text{Número de artículos} + F \) Sustituyendo los valores tenemos: \( \text{Costo total} = 2 \veces 200 + 300 \) \( \text{Costo total} = 400 + 300 \) \( \text{Costo total} = 700 \) Por lo tanto, el costo de producir 200 artículos sería de $700.

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