This exercise uses the radioactive decay model. The half-life of strontium-90 is 29 years. How long (in yr) will it take a 70-milligram sample to decay to a mass of 56 mg? (Round your answer to the nearest whole number.) _____ yr

Question

Bytelearn · Accepted Answer

Understand half-life concept: Understand the half-life concept and set up the decay formula. The half-life of a substance is the time it takes for half of the substance to decay. For strontium-90, the half-life is 29 years. The decay formula for a substance with a half-life (h) is given by: $$ y = a \left(\frac{1}{2}\right)^{\frac{x}{h}} $$ where: - $ y $ is the final amount of the substance, - $ a $ is the initial amount of the substance, - $ x $ is the time elapsed, - $ h $ is the half-life of the substance.Insert known values: Insert the known values into the decay formula. We know that the initial amount $ a $ is 70 mg, the final amount $ y $ we want is 56 mg, and the half-life $ h $ is 29 years. We want to find $ x $, the time it takes to decay from 70 mg to 56 mg. So we set up the equation: $$ 56 = 70 \left(\frac{1}{2}\right)^{\frac{x}{29}} $$Solve for x: Solve for $ x $. First, divide both sides of the equation by 70 to isolate the exponential part: $$ \frac{56}{70} = \left(\frac{1}{2}\right)^{\frac{x}{29}} $$ Simplify the left side of the equation: $$ \frac{4}{5} = \left(\frac{1}{2}\right)^{\frac{x}{29}} $$Take logarithm: Take the logarithm of both sides to solve for $ x $. We can use the natural logarithm (ln) to help solve for $ x $: $$ \ln\left(\frac{4}{5}\right) = \ln\left(\left(\frac{1}{2}\right)^{\frac{x}{29}}\right) $$ Using the power rule of logarithms, we can move the exponent in front of the ln: $$ \ln\left(\frac{4}{5}\right) = \frac{x}{29} \cdot \ln\left(\frac{1}{2}\right) $$Isolate and solve: Isolate $ x $ and solve. Divide both sides by $ \ln\left(\frac{1}{2}\right) $ to get $ x $ by itself: $$ x = \frac{\ln\left(\frac{4}{5}\right)}{\ln\left(\frac{1}{2}\right)} \cdot 29 $$ Now we can use a calculator to find the value of $ x $: $$ x \approx \frac{\ln(0.8)}{\ln(0.5)} \cdot 29 $$ $$ x \approx \frac{-0.22314}{-0.69315} \cdot 29 $$ $$ x \approx 0.32193 \cdot 29 $$ $$ x \approx 9.336 $$ Round the answer to the nearest whole number: $$ x \approx 9 $$

This exercise uses the radioactive decay model. $\newline$ The half-life of strontium $-90$ is $29$ years. How long (in yr) will it take a $70$ -milligram sample to decay to a mass of $56$ mg? (Round your answer to the nearest whole number.) $\newline$ $\_\_\_\_\_$ yr

Full solution

More problems from Write exponential functions: word problems

Related topics

This exercise uses the radioactive decay model.\newlineThe half-life of strontium−90-90−90 is 292929 years. How long (in yr) will it take a 707070-milligram sample to decay to a mass of 565656 mg? (Round your answer to the nearest whole number.)\newline_____\_\_\_\_\______yr

This exercise uses the radioactive decay model. $\newline$ The half-life of strontium $-90$ is $29$ years. How long (in yr) will it take a $70$ -milligram sample to decay to a mass of $56$ mg? (Round your answer to the nearest whole number.) $\newline$ $\_\_\_\_\_$ yr