a scientist has 100 milligrams of a radioactive element. the amount of radioactive element remaining after t…

a scientist has 100 milligrams of a radioactive element. the amount of radioactive element remaining after t days can be determined using the equation (f(t)=100(\frac{1}{2})^{\frac{t}{10}}). after three days, the scientist receives a second shipment of 100 milligrams of the same element. the equation used to represent the amount of shipment 2 remaining after t days is (f(t)=100(\frac{1}{2})^{\frac{t - 3}{10}}). after any time, t, the mass of the element remaining in shipment 1 is what percentage of the mass of the element remaining in shipment 2?\n78.1%\n81.2%\n123.1%\n128.0%

a scientist has 100 milligrams of a radioactive element. the amount of radioactive element remaining after t days can be determined using the equation (f(t)=100(\frac{1}{2})^{\frac{t}{10}}). after three days, the scientist receives a second shipment of 100 milligrams of the same element. the equation used to represent the amount of shipment 2 remaining after t days is (f(t)=100(\frac{1}{2})^{\frac{t - 3}{10}}). after any time, t, the mass of the element remaining in shipment 1 is what percentage of the mass of the element remaining in shipment 2?\n78.1%\n81.2%\n123.1%\n128.0%

Answer

Answer:

A. 78.1%

Explanation:

Step1: Write the mass - equations for both shipments

Let $m_1(t)=100\left(\frac{1}{2}\right)^{\frac{t}{10}}$ be the mass of shipment 1 and $m_2(t)=100\left(\frac{1}{2}\right)^{\frac{t - 3}{10}}$ be the mass of shipment 2 at time $t$.

Step2: Calculate the ratio of $m_1(t)$ to $m_2(t)$

The ratio $r=\frac{m_1(t)}{m_2(t)}=\frac{100\left(\frac{1}{2}\right)^{\frac{t}{10}}}{100\left(\frac{1}{2}\right)^{\frac{t - 3}{10}}}$. Using the rule of exponents $\frac{a^m}{a^n}=a^{m - n}$, we have $r=\left(\frac{1}{2}\right)^{\frac{t}{10}-\frac{t - 3}{10}}$.

Step3: Simplify the exponent

$\frac{t}{10}-\frac{t - 3}{10}=\frac{t-(t - 3)}{10}=\frac{t - t+3}{10}=\frac{3}{10}$. So $r=\left(\frac{1}{2}\right)^{\frac{3}{10}}$.

Step4: Convert the ratio to a percentage

$r=\left(\frac{1}{2}\right)^{\frac{3}{10}}=2^{-\frac{3}{10}}\approx0.781$. To convert to a percentage, we multiply by 100, getting $0.781\times100 = 78.1%$.