{"status": "success", "data": {"description_md": "\nArgon-$39$, one of the four unstable Argon isotopes, has a half-life of $269$ years. This means that after $269$ years, only half of the initial Argon-$39$ remains, decaying exponentially.\n\nThe decay is modeled by the equation $P(t) = \\left(\\frac{1}{2}\\right)^{\\frac{t}{t_0}}$, where $P(t)$ is the remaining portion of Argon-$39$ after $t$ years, and $t_0$ is the half-life.\n\nGiven that $69\\%$ of the Argon-$39$ remains in a fossil, determine $\\lfloor X \\rfloor$, where $X$ is the age of the fossil in years.", "description_html": "

Argon-39, one of the four unstable Argon isotopes, has a half-life of 269 years. This means that after 269 years, only half of the initial Argon-39 remains, decaying exponentially.

The decay is modeled by the equation P(t) = \\left(\\frac{1}{2}\\right)^{\\frac{t}{t_0}}, where P(t) is the remaining portion of Argon-39 after t years, and t_0 is the half-life.

Given that 69\\% of the Argon-39 remains in a fossil, determine \\lfloor X \\rfloor, where X is the age of the fossil in years.

", "hints_md": "1. The \"full-life\" (which fun fact: does not in fact exist if analyzed exponentially) is **NOT** $538$ years. Isotopes do not decay in a linear function.\n2. This question aims to introduce you to the basics of logarithimic functions. Aim to utilize a logarithmic function somewhere when solving the problem.", "hints_html": "
  1. The “full-life” (which fun fact: does not in fact exist if analyzed exponentially) is NOT 538 years. Isotopes do not decay in a linear function.
  2. This question aims to introduce you to the basics of logarithimic functions. Aim to utilize a logarithmic function somewhere when solving the problem.
", "editorial_md": "We can substitute the values given into the Half-Life formula in the following method.\n\n$$\n\\begin{aligned}\n 0.69 &= \\left(\\frac{1}{2}\\right)^{\\frac{X}{269}} \\\\\n \\frac{X}{269} &= \\frac{\\log 0.69}{\\log 0.5} \\\\\n X &\\approx 144.00 \\\\\n \\lfloor X \\rfloor &= 144\n\\end{aligned}\n$$\n\n$\\therefore$$\\lfloor X \\rfloor$ is equal to $144$ .\n\n - htoshiro\n", "editorial_html": "

We can substitute the values given into the Half-Life formula in the following method.

\\begin{aligned} 0.69 &= \\left(\\frac{1}{2}\\right)^{\\frac{X}{269}} \\\\ \\frac{X}{269} &= \\frac{\\log 0.69}{\\log 0.5} \\\\ X &\\approx 144.00 \\\\ \\lfloor X \\rfloor &= 144 \\end{aligned}

\\therefore\\lfloor X \\rfloor is equal to 144 .

", "flag_hint": "", "point_value": 2, "problem_name": "Half-Life?", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}