strontium - 90 is a radioactive material that decays according to the function ( a(t)=a_{0}e^{-0.0244t} )…

strontium - 90 is a radioactive material that decays according to the function ( a(t)=a_{0}e^{-0.0244t} ), where ( a_{0} ) is the initial amount present and ( a ) is the amount present at time ( t ) (in years). assume that a scientist has a sample of 800 grams of strontium - 90.\n(a) what is the decay rate of strontium - 90?\n(b) how much strontium - 90 is left after 40 years?\n(c) when will only 600 grams of strontium - 90 be left?\n(d) what is the half - life of strontium - 90?\n(a) the decay rate of strontium - 90 is (square%)\n(type an integer or a decimal. include the negative sign for the decay rate.)\n(b) approximately (square) grams of strontium - 90 is left after 40 years.\n(do not round until the final answer. then round to the nearest whole number as needed.)\n(c) only 600 grams of strontium - 90 will be left in about (square) years.\n(do not round until the final answer. then round to the nearest tenth as needed.)\n(d) the half - life of strontium - 90 is approximately (square) years.\n(do not round until the final answer. then round to the nearest tenth as needed.)

strontium - 90 is a radioactive material that decays according to the function ( a(t)=a_{0}e^{-0.0244t} ), where ( a_{0} ) is the initial amount present and ( a ) is the amount present at time ( t ) (in years). assume that a scientist has a sample of 800 grams of strontium - 90.\n(a) what is the decay rate of strontium - 90?\n(b) how much strontium - 90 is left after 40 years?\n(c) when will only 600 grams of strontium - 90 be left?\n(d) what is the half - life of strontium - 90?\n(a) the decay rate of strontium - 90 is (square%)\n(type an integer or a decimal. include the negative sign for the decay rate.)\n(b) approximately (square) grams of strontium - 90 is left after 40 years.\n(do not round until the final answer. then round to the nearest whole number as needed.)\n(c) only 600 grams of strontium - 90 will be left in about (square) years.\n(do not round until the final answer. then round to the nearest tenth as needed.)\n(d) the half - life of strontium - 90 is approximately (square) years.\n(do not round until the final answer. then round to the nearest tenth as needed.)

Answer

Explanation:

Step1: Find the decay rate

The general form of exponential decay is (A(t)=A_0e^{rt}), where (r) is the decay rate. Given (A(t)=A_0e^{- 0.0244t}), so (r=-0.0244). To convert to percentage, multiply by (100): (-0.0244\times100=-2.44%)

Step2: Find the amount after 40 years

Given (A_0 = 800), (t = 40), and (A(t)=A_0e^{-0.0244t}) Substitute the values: (A(40)=800e^{-0.0244\times40}) First calculate the exponent: (-0.0244\times40=-0.976) Then (A(40)=800e^{- 0.976}) Using (e^{-0.976}\approx0.3769) (A(40)=800\times0.3769 = 301.52\approx302)

Step3: Find when (A(t) = 600)

Set (A(t)=600), (A_0 = 800) in (A(t)=A_0e^{-0.0244t}) (600 = 800e^{-0.0244t}) Divide both sides by (800): (\frac{600}{800}=e^{-0.0244t}) (0.75=e^{-0.0244t}) Take the natural logarithm of both sides: (\ln(0.75)=-0.0244t) (t=\frac{\ln(0.75)}{-0.0244}) Since (\ln(0.75)\approx - 0.2877) (t=\frac{-0.2877}{-0.0244}\approx11.8)

Step4: Find the half - life

For half - life, (A(t)=\frac{A_0}{2}) Set (A(t)=\frac{A_0}{2}) in (A(t)=A_0e^{-0.0244t}) (\frac{A_0}{2}=A_0e^{-0.0244t}) Divide both sides by (A_0): (\frac{1}{2}=e^{-0.0244t}) Take the natural logarithm of both sides: (\ln(\frac{1}{2})=-0.0244t) Since (\ln(\frac{1}{2})=-\ln(2)\approx - 0.6931) (t=\frac{-\ln(2)}{-0.0244}=\frac{0.6931}{0.0244}\approx28.4)

Answer:

(a) (-2.44) (b) (302) (c) (11.8) (d) (28.4)