Question

you have 850 grams of an an unknown radioactive substance that has been determined to decay according to (d(t)=850e^{-0.001487cdot t}) where (t) is in years. according to this equation, how much is the initial (starting amount)? grams how long before half of this initial amount has decayed? round to the nearest year. years

Explanation:

Step1: Find the initial amount

The initial amount is when $t = 0$. Substitute $t=0$ into $D(t)=850e^{- 0.001487t}$. Since $e^0 = 1$, we have $D(0)=850e^{-0.001487\times0}=850\times1 = 850$ grams.

Step2: Find the half - life

We want to find $t$ when $D(t)=\frac{850}{2}=425$. Set up the equation $425 = 850e^{-0.001487t}$. First, divide both sides by 850: $\frac{425}{850}=e^{-0.001487t}$, so $0.5 = e^{-0.001487t}$. Take the natural logarithm of both sides: $\ln(0.5)=\ln(e^{-0.001487t})$. Using the property $\ln(e^x)=x$, we get $\ln(0.5)=- 0.001487t$. Then $t=\frac{\ln(0.5)}{-0.001487}$. Calculate $\ln(0.5)\approx - 0.693147$, so $t=\frac{-0.693147}{-0.001487}\approx466$ years.

Answer:

850 grams
466 years