FACULTAD DE INGENIERÍA CURSO: CÁLCULO DE VARIAS VARIABLES PROFESORA: ZELA APAZA, ANA MARIA
PROYECTO FORMATIVO EN MATEMÁTICA EXTRACCIÓN MINERA EN EL PERÚ INTEGRANTES: 1. 2. 3. 4. 5. 6. 7.
De la Torre Picho, Gabriel E. Dioses Salinas, Diana C. Flores Villanueva, Annelisie Z Ghiorzo Avalos, Daniella A. Guivin Zamora, Santiago Navarro Yaya, María G. Sánchez Sullca, Daniel.
FECHA: 31/10/2016 BLOQUE: FC-PREAGR03A1T 1
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Índice Página 1. Contexto Con texto de la minería 1.1 Introducción
3
1.2 Resumen
5
1.3 Abstract
6
1.3 Contextualización y definiciones
7
2. Etapas 2.1 Primera Etapa: Situaciones problemáticas
10
2.1.1 Situación problemática 1
10
2.1.2 Situación problemática 2
15
2.2 Segunda Etapa: Estudio de casos
20
2.2.1 Caso 1
20
2.2.2 Caso 2
26
2.3 Tercera Etapa: Responsabilidad social y contaminación ambiental 3. Bibliografía
38 49
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INTRODUCCIÓN: La minería es una actividad económica extractiva, contextualizado por la explotación de minerales, que se encuentran en los suelos y/o subsuelos en forma de yacimientos. Se sabe que esta actividad es una de las más antiguas de la humanidad, ya que data desde tiempos de la prehistoria, que se utilizaban para la fabricación de armas y de herramientas para el ser humano. Con el pasar del tiempo se convierto una actividad importante en el sector industrial, creando diversas técnicas que van desde mejorando las cualidades químicas- físicas del mineral hasta investigar la mejor forma de obtenerlos y explotarlos. Actualmente las empresas mineras o compañías son las encargas de llevarlo a cabo como industria que se divide en gran, mediana y pequeña minería, no obstante, en algunos países existe una cuarta categoría, la artesanal. Por la gran demanda de esta actividad, ya que es uno de los pilares en la economía mundial según el Banco Mundial, muchos países con el pasar de los años vieron, que esta actividad debe ser supervisada así que crearon leyes y legislaciones que buscan principalmente la armonía y estabilidad de esta actividad productiva enfocándose en la seguridad de los mineros y el cuidado del medio ambiente principalmente. Lamentablemente las violaciones a estas leyes, por ejemplo: la minería ilegal o la minería informal causan un gran desastre ecológico que en los últimos tiempos, ha causado diversos desastres
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En nuestro país trae muchos beneficios, ya que en el Perú es uno de los p ilares en la economía representando un 15% PBI nacional y el 60% de las exportaciones totales que tiene nuestro país.
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RESUMEN: El siguiente proyecto propone emplear el cálculo con varias variables para el planteamiento y resolución de problemas que surgen durante el proceso de extracción minera utilizando términos como las curvas de nivel, derivadas parciales e integrales dobles y sesgos económicos y ambientales por medio del uso de software y cálculos tradicionales, puesto que son de suma importancia en el mundo actual.
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ABSTRACT: The following project develops the way we use calculus of various variables to establish and solve problems involving in the mining and extraction process by using terms like contours, partials and double integrals and economic and environmental features, since they are utterly important to the world today.
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CONTEXTUALIZACIÓN DEL PROBLEMA El Perú es un país de antigua tradición minera, tradición qu e mantiene y cultiva gracias a la presencia de empresas líderes a nivel internacional. Contamos con un enorme potencial geológico, la presencia de la Cordillera de los Andes a lo largo del territorio, constituye nuestra principal fuente de recursos minerales. A nivel mundial y latinoamericano el Perú se ubica entre los primeros productores de diversos metales, (oro, plata, cobre, plomo, zinc, hierro, estaño, molibdeno, teluro, entre otros), lo cual es reflejo no sólo de la abundancia de recursos y la capacidad de producción de la actividad minera peruana, sino de la estabilidad de las políticas económicas en nuestro país. Los minerales producidos en el Perú son de gran demanda en el mercado mundial actual, cuyo desarrollo se basa en la producción y la industria. Estados Unidos, China, Suiza, Japón, Canadá y la Unión Europea son los principales demandantes. Las oportunidades que ofrece nuestro país han hecho que seamos uno de los países de la región donde se observa mayor inversión en minería, con
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cartera estimada de estos proyectos mineros destaca inversiones muy importantes en Cuzco, Cajamarca, Moquegua, Tacna, Ica, Lima, Arequipa, Junín, Piura, Ancash, Huancavelica y Lambayeque.
DEFINICIÓN: La minería es la extracción selectiva de minerales y otros materiales (salvo materiales orgánicos de formación reciente) a partir de la corteza terrestre. Se encuentran en el suelo y/o subsuelo subsu elo en forma de yacimientos. También También es una de las actividades más antiguas de la humanidad. Casi desde el principio de la edad de piedra, hace 2,5 millones de años o más. La minería siempre implica la extracción física de materiales con frecuencia en grandes cantidades para recuperar solo pequeños volúmenes del material deseado (oro, plata, entre otros.). Por eso resulta imposible que no afecte al medio ambiente, en la zona de la mina.
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No metálicos o de cantera y construcción: Incluye la arena, la grava, los áridos, las arcillas para ladrillos, la caliza y los esquistos para la fabricación de cemento. En este grupo también se incluyen la pizarra para tejados y las piedras pulidas, como la caliza, el granito, los travertinos o el mármol.
Industriales: incluye el cuarzo, la trona, la sal común, el potasio, el Amianto, el talco, el feldespato, el azufre azufre y los fosfatos.
Gemas: Incluyen los diamantes, los rubíes, los zafiros y las esmeraldas.
Combustibles: incluyen el carbón, el lignito, la turba, el petróleo y el gas (aunque generalmente estos últimos no se consideran productos mineros). El uranio se incluye con frecuencia entre los combustibles.
IMPORTANCIA: La minería como ya antes mencionada, es uno de los pilares de la economía especialmente para países que su mayor ingreso son las actividades extracticos, ya que generan una fuente ocupacional de gran importancia,
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PRIMERA ETAPA: SITUACIONES PROBLEMÁTICAS. SITUACIÓN PROBLEMÁTICA 1 La productividad de una empresa dedicada a la fabricación de equipos de seguridad personal para trabajadores de una mina, está dada por:
;; =90.25//
Unidades, donde L indica la fuerza laboral en miles horas-trabajador y K es es el capital en unidades de millón de dólares.
a) Calcule la productividad marginal del capital y la productividad productividad marginal de la mano de obra cuando el capital es de 5,495 miles de millones de dólares y el nivel de la fuerza laboral es 4 785 000 horas-trabajador.
Para hallar la productividad marginal se debe desarrollar la derivada parcial en función de la variable de la cual se quiere la productividad y evaluar con los valores otorgados (5, 495,000; 4, 785,000). Las derivadas parciales resultan:
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=62.9 b) ¿La gerencia de la empresa empresa debe estimular la inversión inversión de capital o el incremento de la fuerza laboral para aumentar la productividad tan rápidamente como sea posible?
En el incremento de la fuerza laboral, ya que puesto que su productividad marginal es mayor, eso indica que el crecimiento en la fuerza laboral harán un incremento en la productividad mayor que un incremento de la misma magnitud en el capital.
c) Con la ayuda de un software, dibuje el mapa de contorno para Q = 100; 140; 180; 220 y 250 . ¿Qué relación entre K y L se puede deducir del
mapa de contorno?
Utilizando el programa Fooplot , nos encontramos con las siguientes curvas tras despejar Y en la ecuación:
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Cuadro 1
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. = ./ =
Podemos observar que cuando el límite de la función a
∞
se aproxima
, la función se acerca a 0. Por otro lado, cuando el límite de la
0,265 = 0 =0,265
función se aproxima a debe a que
y
, la función crece indefinidamente. Esto se son asíntotas.
d) Use el mapa de contorno para estimar la producción para el par (L, K) = (145,198).
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Satisface: Cuadro 2
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SITUACIÓN PROBLEMÁTICA 2 Las Bambas es una mina de cobre de gran envergadura, se ubica a más de 4,000 m.s.n.m., entre las provincias de Cotabambas y Grau, Región Apurímac, a 70 kilómetros, en línea recta, al sudeste de la ciudad de Abancay. Produce concentrado de cobre (subproductos de oro y plata) y concentrado de molibdeno. En el mapa adjunto se muestran algunas líneas de contorno de la función altura, en los alrededores del centro minero. Cuadro 3
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a) Ustedes pueden apreciar que que en algunas zonas de la figura adjunta las curvas de nivel están cerca unas de otras y en otras zonas las curvas de nivel están alejadas entre sí. ¿Qué significa esto en términos del relieve de la región? Explique y justifique su respuesta.
En la topografía, las las curvas de nivel se obtienen al cortar una superficie con un plano horizontal (
=
y proyectar el cruce en el plano XY. El
valor de c (es decir, altitud) es, mayormente, cada 20, 40, 50, hasta 100m. Por ejemplo, en el gráfico mostrado, cada curva de nivel se realiza a cada 40 metros de altitud (o m.s.n.m.) porque entre cada curva maestra (las curvas en negrita cada 200 m.s.n.m.) se encuentras 4 curvas más.
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b) Si una persona se desplaza a lo largo de cualquier curva de nivel, ¿asciende o desciende? Justifique su respuesta. Ninguno, se mantiene a la misma altitud, porque es eso lo que representa una curva de nivel. Cuadro 4
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c) Use el mapa de contorno, con la escala indicada, indicada, para estimar el valor de la derivada direccional de la función altura en Las Bambas en la dirección al nor-oeste.
Tomando en cuenta la definición de la derivada derivada direccional dada por el vector unitario u, donde el límite existe:
ℎ ℎ
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≈ 42004100 40 ≈ 100 40 ≈2.5
Se puede estimar que la derivada direccional es 2.5m.
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SEGUNDA ETAPA: Estudio de casos CASO 1: Modelo de un inventario con retraso de pedido. La finalidad del modelo de la cantidad económica de pedido (economic order quantity, EOQ) es encontrar la cantidad que debe pedirse de un objeto o producto con el fin de minimizar los costos totales del inventario. El modelo supone tres componentes diferentes del costo: costo de pedido, son los que implican colocar y recibir una orden; costo de manteamiento del inventario o
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a) Modelar las expresiones generales para que q y s que minimizan los costos totales anuales de inventario Hallando puntos críticos:
= −−.−− −− −−.−− −− = 0
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1 + =
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s=
. . √ . .+.
=4997 =4997 unidades.
La escasez máxima, en unidades es 4997 unidades.
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c) La siguiente gráfica muestra un típico ciclo de inventario aplicable al modelo de inventario de pedidos con retraso.
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S= Escasez máxima, en unidades.
q= Cantidad de unidades que se piden, (cantidad de pedido).
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CASO 2: Área de ejecución del proyecto Toromocho El Proyecto Toromocho consiste en una mina de tajo abierto con reservas de
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