Modelado_de_funciones.ipynb

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  "metadata": {
    "colab": {
      "provenance": [],
      "authorship_tag": "ABX9TyMZUSS1dOCCw5zExbzVoT7b",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/Polux79/Ingenieria-ambiental/blob/main/Modelado_de_funciones.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Carga de librerías necesarias\n",
        "from math import log\n",
        "import sympy as sp\n",
        "from sympy import *\n",
        "init_printing()"
      ],
      "metadata": {
        "id": "oh2BQSTH_-W5"
      },
      "execution_count": 47,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "# MODELADO CON FUNCIONES EXPONENCIALES Y LOGARÍTMICAS\n",
        "\n",
        "**Ejemplo 1** .- Crecimiento exponencial\n",
        "\n",
        "Predecir el tamaño de la población de bacterias si la población inicial en el cultivo es $1200$. Luego, se realiza un cuenta muestral de bacterias en el cultivo y se encuentra que la tasa de crecimiento relativo es de $28\\%$ por hora.\n",
        "1. Encuentre una función que modele el número de bacterias después de $t$ horas.\n",
        "2. ¿Cuál es la cuenta estimada después de un día?\n",
        "3. Trace la gráfica de la función $N(t)$"
      ],
      "metadata": {
        "id": "uKMpX8aA0peA"
      }
    },
    {
      "cell_type": "code",
      "execution_count": 48,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 52
        },
        "id": "uX885Qw20ov1",
        "outputId": "9a3ae185-c379-45cc-803d-387a6ec93135"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "d                0.28⋅t\n",
              "──(N(t)) = 1200⋅ℯ      \n",
              "dt                     "
            ],
            "text/latex": "$\\displaystyle \\frac{d}{d t} N{\\left(t \\right)} = 1200 e^{0.28 t}$"
          },
          "metadata": {}
        }
      ],
      "source": [
        "#Declaración de variables\n",
        "t = sp.symbols('t')  # horas\n",
        "y = sp.Function('N') # Cantidad de bacterias\n",
        "\n",
        "Ecuacion = Eq(y(t).diff(), 1200*exp(0.28*t))\n",
        "#Presentación de las ecuaciones resueltas\n",
        "display(Ecuacion)"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Evaluación de N(t) para t = 24 horas, presentación de resultados con 6 cifras significativas\n",
        "display(Ecuacion.subs(t,24).rhs.evalf(6))\n",
        "\n",
        "# Gráfico de la función ---> hasta t = 30 horas\n",
        "sp.plot(Ecuacion.rhs,(t,0,30), line_color='b',ylabel='Cantidad (bacterias)', xlabel='Tiempo (h)')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 523
        },
        "id": "HHVoa-4pMC_V",
        "outputId": "b9e54e04-6f29-48be-929d-09dff8cf67b6"
      },
      "execution_count": 49,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "994581."
            ],
            "text/latex": "$\\displaystyle 994581.0$"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<sympy.plotting.plot.Plot at 0x7e8a2ff1bb20>"
            ]
          },
          "metadata": {},
          "execution_count": 49
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "\n",
        "---\n",
        "\n"
      ],
      "metadata": {
        "id": "tgzSuB03s3Sh"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Ejemplo 2** .- Crecimiento exponencial\n",
        "\n",
        "En el pueblo de Soyocyatamjurpacusitac, la población en 1970 fue de $170000$ habitantes y en 1990 de $210000$. ¿Cuál será la población el año 2010?"
      ],
      "metadata": {
        "id": "1iDLeDMqZPQA"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def calcular_tasa_crecimiento(No, N, t):\n",
        "    try:\n",
        "        r = log(N / No) / t\n",
        "        return round(r, 4)\n",
        "\n",
        "    except ZeroDivisionError:\n",
        "        print(\"Error: El tiempo transcurrido no puede ser cero.\")\n",
        "        return None\n",
        "    except ValueError:\n",
        "        print(\"Error: Ingrese valores numéricos válidos.\")\n",
        "        return None\n",
        "\n",
        "# Ejemplo de uso\n",
        "No = float(input(\"Ingrese el valor inicial de la población: \"))\n",
        "N = float(input(\"Ingrese el valor final de la población: \"))\n",
        "t = float(input(\"Ingrese el tiempo transcurrido entre N y No: \"))\n",
        "\n",
        "tasa_crecimiento = calcular_tasa_crecimiento(No, N, t)\n",
        "\n",
        "if tasa_crecimiento is not None:\n",
        "    print(f\"La tasa de crecimiento es: {tasa_crecimiento}\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "UOELzp87k6m7",
        "outputId": "57a6b1d3-d458-48e7-888e-4db80c22432c"
      },
      "execution_count": 77,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Ingrese el valor inicial de la población: 170000\n",
            "Ingrese el valor final de la población: 210000\n",
            "Ingrese el tiempo transcurrido entre N y No: 20\n",
            "La tasa de crecimiento es: 0.0106\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#Declaración de variables\n",
        "t = sp.symbols('t')  # horas\n",
        "y = sp.Function('N') # Cantidad de pobladores\n",
        "\n",
        "Ecuacion = Eq(y(t).diff(), 170000*exp(tasa_crecimiento*t))\n",
        "\n",
        "#Presentación de la ecuación resuelta\n",
        "display(Ecuacion)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 52
        },
        "id": "m_w3oY3ImGU8",
        "outputId": "5d887c2d-ef63-47cc-d3b1-9b497ec58f2a"
      },
      "execution_count": 80,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "d                  0.0106⋅t\n",
              "──(N(t)) = 170000⋅ℯ        \n",
              "dt                         "
            ],
            "text/latex": "$\\displaystyle \\frac{d}{d t} N{\\left(t \\right)} = 170000 e^{0.0106 t}$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Evaluación de N(t) para t = 40 años, presentación de resultados con 6 cifras significativas\n",
        "display(Ecuacion.subs(t,40).rhs.evalf(6))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        },
        "id": "ED_1GQOursb9",
        "outputId": "2bd9f549-0566-4ccf-b5a3-e6080b589cbb"
      },
      "execution_count": 93,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "259744."
            ],
            "text/latex": "$\\displaystyle 259744.0$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "\n",
        "---\n",
        "\n"
      ],
      "metadata": {
        "id": "aQrQ0c6Os6j4"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Ejemplo 3** .- Decaimiento radiactivo\n",
        "\n",
        "Utilizado en química, para determinar la cantidad de material radiactivo que va quedando (decaimiento espontáneo) conforme transcurre el tiempo. La tasa de decaimiento es directamente proporcional a la masa de la sustancia, es decir, la masa de material reactivo disminuye. Sea $N(t)$, la cantidad de sustancia radiactiva ($masa$) que permanece en el tiempo $t$, $r$ es la tasa de decaimiento como una proporción de la masa, se calculo como el cociente del $ln $ $2$ dividido entre la *vida media* del material reactivo $t_{1/2}$, es decir, $r = \\frac{ln 2}{t_{1/2}}$ y $N_o$ es una constante que indica la cantidad de material radiactivo inicial en el $t = 0$, se cumple, $N(0) = N_o$, el signo negativo representa la disminución de la cantidad $N$, conforme transcurre el tiempo. La masa experimenta un decaimiento exponencial, disminuye según el modelo:   $N(t) = N_oe^{-rt}$\n"
      ],
      "metadata": {
        "id": "k-rqCRgos-_x"
      }
    },
    {
      "cell_type": "code",
      "source": [],
      "metadata": {
        "id": "aZVB7DvnwmpA"
      },
      "execution_count": null,
      "outputs": []
    }
  ]
}