Answer:
3. The model for a substance of 400 mg with a half-life of 84 days is:
[tex]y(t)=400e^{-0.00825\cdot t}[/tex]
4. 160 mg.
5. It will take 7 years for Ahmed's account to accumulate $500.
Step-by-step explanation:
3. We can write a model for the substance with exponential decay as:
[tex]y=Ce^{mt}[/tex]
We know that at time t=0, the mass is 400 mg.
[tex]y(0)=Ce^{m\cdot0}=C\cdot 1=400\\\\C=400[/tex]
We also know that the half-life is 84 days, so at t=84 days, the mass will be 400/2=200 mg.
[tex]y(84)=400e^{m\cdot84}=200\\\\e^{84m}=200/400=0.5\\\\84m=ln(0.5)\\\\m=ln(0.5)/84=-0.00825[/tex]
The model for a substance of 400 mg with a half-life of 84 days is:
[tex]y(t)=400e^{-0.00825\cdot t}[/tex]
4. After 110 days the substance will have a mass of y=160 mg.
[tex]y(110)=400e^{-0.00825\cdot 110}=400e^{-0.9}=400*0.4=160[/tex]
5. The amount that Ahmed's will acumulate in function of the years (n) can be written as:
[tex]C(n)=400\cdot(1+0.035)^n=400\cdot1.035^n[/tex]
We can calculate how many years it will take for the Amhed's account to acumulate $500 as:
[tex]C(n)=500=400\cdot 1.035^n\\\\1.035^n=500/400=1.25\\\\n\cdot ln(1.035)=ln(1.25)\\\\n=ln(1.25)/ln(1.035)=0.223/0.034=6.49\approx7[/tex]
It will take 7 years for Ahmed's account to accumulate $500.
