radioactive life

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$3500 are invested in an account that pays interest of the annual rate of 4%. Determine equation that gives balance in the amount t years after inception of account. f(t)=3500(1.04)t

Half-life of radioactive substance is 300 years. Fill in the following table if initial amount is 900 lb.

t f(t)
0 900
300 450
600 225

This is what I got so far:

ab ^0=900 ----> a=900
ab ^300=450 ----> 900b ^300/900=450/900
b ^300=.5 x 1/300

Am I on the right track?
 
Hello, Angela!

In the half-life problem, you were fine up to the last step . . .

Last step: \(\displaystyle \;b\:=\:(0.5)^{\frac{1}{300}}\:\approx\:0.99769\)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Are you supposed be using "e" in this problem?

Then:   f(t)=aekt\displaystyle \;f(t)\:=\:a\cdot e^{-kt}

Since f(0)=900:    ae0=900        a=900\displaystyle f(0)\,=\,900:\;\;a\cdot e^0\,=\,900\;\;\Rightarrow\;\;a\,=\,900

. . and we have:   f(t)=900ekt\displaystyle \;f(t)\:=\:900\cdot e^{-kt}


Since f(300)=40:    900e300k=450        e300k=0.5\displaystyle f(300)\,=\,40:\;\;900\cdot e^{-300k}\,=\,450\;\;\Rightarrow\;\;e^{-300k}\,=\,0.5

. . -300k=ln(0.5)        k=ln(0.5)300  0.00231\displaystyle 300k\,=\,\ln(0.5)\;\;\Rightarrow\;\;k\,=\,\frac{\ln(0.5)}{-300}\:\approx\;0.00231

Therefore:   f(t)=900e0.00231t\displaystyle \;f(t)\:=\:900\cdot e^{-0.00231t}
 
How do you do that on the calculator, because I got .00166 when I did (.5) x (1/300).
 
Hello, angelasloan7038!

How do you do that on the calculator, because I got .00166 when I did (.5) x (1/300).
You're multiplying!

It says: (0.5)1300    That’s 0.5 to the 1300  power!\displaystyle \text{It says: }(0.5)^{\frac{1}{300}}\;\cdots\; \text{That's }0.5\text{ to the }\frac{1}{300}\;power!



Do you know that 72\displaystyle 7^2 does not mean 7×2\displaystyle 7\,\times\,2 ?

. . . . . . . . . . .that 312\displaystyle 3^{\frac{1}{2}} does not mean 3×12\displaystyle 3\,\times\,\frac{1}{2} ?
 
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