Element X decays radioactively with a half life of 5 minutes. If there are 890 grams of Element X , how long, to the nearest tenth of a minute, would it take the element to decay to 194 grams? \[ y=a(.5)^{\frac{t}{h}} \]

To find out how long it takes for 890 grams of Element X to decay to 194 grams, you can use the provided formula. Here, \(y\) represents the remaining amount, \(a\) is the initial amount (890 grams), \(h\) is the half-life (5 minutes), and \(t\) is the time elapsed. We need to set up the equation: \[194 = 890(0.5)^{\frac{t}{5}}\] Now, divide both sides by 890 to isolate the exponential term: \[\frac{194}{890} = (0.5)^{\frac{t}{5}}\] This simplifies to approximately 0.218. Now, take the log of both sides: \[\log(0.218) = \frac{t}{5} \cdot \log(0.5)\] Solving for \(t\), we find: \[t = 5 \cdot \frac{\log(0.218)}{\log(0.5)}\] Calculating this will give you the time it takes for Element X to decay to 194 grams. By closely following these calculations, you will end up with \(t \approx 10.4\) minutes when rounded to the nearest tenth! For you math enthusiasts, exponential decay isn't just a mathematical concept—it's the underlying principle for dating ancient artifacts! Techniques like radiocarbon dating, which rely on the natural decay of carbon-14, help researchers determine the age of organic materials, providing a fascinating glimpse into our past. Who knew half-lives could hold the key to unraveling history? If you're keen to dive deeper into this subject, consider picking up a book on nuclear chemistry or the principles of radioactive decay. Titles like "The Physics of Atomic Nuclei" or "Introduction to Radiological Physics and Radiation Dosimetry" offer accessible insights that blend complexity with clarity. You won’t just be reading—you'll be journeying into the very fabric of atoms!