element x decays radioactively with a half life of 13 minutes. if there are 860 grams of element x, how…

element x decays radioactively with a half life of 13 minutes. if there are 860 grams of element x, how long, to the nearest tenth of a minute, would it take the element to decay to 152 grams?\n$y = a(.5)^{\frac{t}{h}}$

element x decays radioactively with a half life of 13 minutes. if there are 860 grams of element x, how long, to the nearest tenth of a minute, would it take the element to decay to 152 grams?\n$y = a(.5)^{\frac{t}{h}}$

Answer

Explanation:

Step1: Identify given values

$a=860$, $y=152$, $h=13$

Step2: Substitute into decay formula

$$152 = 860(0.5)^{\frac{t}{13}}$$

Step3: Isolate the exponential term

$$\frac{152}{860} = (0.5)^{\frac{t}{13}}$$ Simplify left side: $\frac{152}{860} \approx 0.1767$

Step4: Take log of both sides

$$\log(0.1767) = \log\left((0.5)^{\frac{t}{13}}\right)$$ Use log power rule: $\log(0.1767) = \frac{t}{13}\log(0.5)$

Step5: Solve for t

$$t = 13 \times \frac{\log(0.1767)}{\log(0.5)}$$ Calculate: $\frac{\log(0.1767)}{\log(0.5)} \approx \frac{-0.752}{-0.3010} \approx 2.498$ $$t \approx 13 \times 2.498 \approx 32.5$$

Answer:

32.5 minutes