Question

Analyze the following situation Juan is starting a new business, he indicates that the price of his product corresponds to p=6000−4x , where x represent the number of tons produced and sold and p It is given in dollars. According to the previous information, what is the maximum income that Juan can obtain with his new product?

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Answer to a math question Analyze the following situation Juan is starting a new business, he indicates that the price of his product corresponds to p=6000−4x , where x represent the number of tons produced and sold and p It is given in dollars. According to the previous information, what is the maximum income that Juan can obtain with his new product?

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Hester
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Los ingresos generados por la venta de un producto son el producto del precio unitario por la cantidad vendida. En este caso, la función de precios de Juan está dada por \( p = 6000 - 4x \), donde \( x \) representa el número de toneladas producidas y vendidas. El ingreso generado al vender \( x \) toneladas a un precio de \( p \) dólares por tonelada está dado por \( \text = p \times x \). Sustituyendo la función de precio \( p = 6000 - 4x \) en la fórmula del ingreso, obtenemos: \[ \texto = (6000 - 4x) \veces x = 6000x - 4x^2 \] Para encontrar el ingreso máximo que Juan puede obtener con su nuevo producto, podemos analizar esta ecuación y determinar el valor de \( x \) que maximiza el ingreso. Esta es una ecuación cuadrática en términos de \( x \), y el valor máximo de la función cuadrática ocurre en su vértice. El vértice de una función cuadrática en la forma \( ax^2 bx c \) está dado por la coordenada x \( x = -\frac \). En este caso, la ecuación cuadrática que representa el ingreso es \( \text = 6000x - 4x^2 \), por lo que comparándola con \( ax^2 bx c \), tenemos \( a = -4 \) y \( b = 6000 \). La coordenada x del vértice es \( x = -\frac = -\frac = \frac = 750 \). Para encontrar el ingreso máximo que Juan puede obtener, sustituye \( x = 750 \) en la función de ingreso: \[ \texto = 6000x - 4x^2 = 6000(750) - 4(750)^2 \] \[ \texto = 4500000 - 2250000 = 2250000 \] Por tanto, el ingreso máximo que Juan puede obtener con su nuevo producto es de $2.250.000.

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