Riddler express 2022-08-19

2022-08-19-express.ipynb

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  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "c4fdd441",
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   "source": [
    "from sympy.solvers import solve\n",
    "from sympy import *"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a8185c7f",
   "metadata": {},
   "source": [
    "`r` is the interest rate (as a percentage) we are solving for"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "90b7bc90",
   "metadata": {},
   "outputs": [],
   "source": [
    "r = Symbol('r')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d970d768",
   "metadata": {},
   "source": [
    "We substitute `72/r` for `t` in the exponential growth equation. On the right side, we have 2, so we subtract it (SymPy supposes all equations are equaled to 0)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "290f5161",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(\\frac{r}{100} + 1\\right)^{\\frac{72}{r}} - 2$"
      ],
      "text/plain": [
       "(r/100 + 1)**(72/r) - 2"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq = (1+r/100)**(72/r)-2\n",
    "eq"
   ]
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   "cell_type": "markdown",
   "id": "0c66d352",
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   "source": [
    "Solving yields a list containing a single solution"
   ]
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   "cell_type": "code",
   "execution_count": 27,
   "id": "acf6d1c8",
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   "outputs": [
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     "name": "stdout",
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     "text": [
      "[-100 - 72*LambertW(-25*2**(11/18)*log(2)/72, -1)/log(2)]\n"
     ]
    }
   ],
   "source": [
    "solutions = solve(eq, r)\n",
    "print(solutions)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "64460f55",
   "metadata": {},
   "source": [
    "Because our variable `r` is found in both the base and power of the equation, our algerbraic solution is quite messy, containing the [Lambert W function](https://en.wikipedia.org/wiki/Lambert_W_function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "67aa9adb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -100 - \\frac{72 W_{-1}\\left(- \\frac{25 \\cdot 2^{\\frac{11}{18}} \\log{\\left(2 \\right)}}{72}\\right)}{\\log{\\left(2 \\right)}}$"
      ],
      "text/plain": [
       "-100 - 72*LambertW(-25*2**(11/18)*log(2)/72, -1)/log(2)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solution = solutions[0]\n",
    "solution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e6b33463",
   "metadata": {},
   "source": [
    "Let's evalutate the numerical solution for the interest rate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "11b5c1e8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 7.84687145301538$"
      ],
      "text/plain": [
       "7.84687145301538"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "interest_rate = solution.evalf()\n",
    "interest_rate"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6cbd55f",
   "metadata": {},
   "source": [
    "Substituting it into the \"Rule of 72\" approximation gets us the approximate doubling time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "070bbbf1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 9.17563138775925$"
      ],
      "text/plain": [
       "9.17563138775925"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rule_72_doubling_time = 72 / interest_rate\n",
    "rule_72_doubling_time"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "62b691d5",
   "metadata": {},
   "source": [
    "Lastly, we check our answer using the interest rate, and doubling time from above. We show it result in 2.0, and therefore, is correct!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "c77b1ede",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2.0$"
      ],
      "text/plain": [
       "2.00000000000000"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1 + interest_rate/100)**rule_72_doubling_time"
   ]
  }
 ],
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