Question

I am having a hard time figuring this out can some one please help!1

As a scientist, you study 200 grams of radioactive material for several months and you have obtained the following data:

Month, Weight

0, 500
2, 329
4, 216.1
6, 142
8, 93
10, 61.29
12, 40.3
Express the weight as the function of months.

Answer 1

`y=500\cdot 0.658^{(x)/(2)}`

Explanation:

Express the weight as the exponential function of months

`y=a\cdot b^{(x)/(2)}`

When
`x=0,` then

`500=a\cdot b^{(0)/(2)}\\ \\500=a\cdot b^0\\ \\500=a`

Hence,

`y=500\cdot b^{(x)/(2)}`

When
`x=2,` then

`329=500\cdot b^{(2)/(2)}\\ \\329=500b^1\\ \\329=500b\\ \\b=(329)/(500)=0.658`

Therefore,

`y=500\cdot 0.658^{(x)/(2)}`

Check the rest values:

`x=4: \ y=500\cdot 0.658^{(4)/(2)}=500\cdot 0.658^2\approx 216\\ \\x=6:\ y=500\cdot 0.658^{(6)/(2)}=500\cdot 0.658^3\approx 142\\ \\x=8:\ y=500\cdot 0.658^{(8)/(2)}=500\cdot 0.658^4\approx 93\\ \\x=10:\ y=500\cdot 0.658^{(10)/(2)}=500\cdot 0.658^5\approx 61\\ \\x=12:\ y=500\cdot 0.658^{(12)/(2)}=500\cdot 0.658^6\approx 40`