Question

The half life of a certain substance is about 4 hours. The graph shows the decay of a 50 gram sample of the substance that is measured every hour for 9 hours.

Which function can be used to determine the approximate number of grams of the sample remaining after t hours?

a
y = 50(0.85)x

b
y = 25(0.15)x

c
y = 50(0.15)x

d
y = 25(0.85)x

Answer 1

I can't see how any of those formulas show exponential decay. (Did you type those correctly?)

The formula that will show the remaining amount correctly is:
ending amount = Bgng Amount / 2^n
where "n" is the number of haf-lives.

So, for example if half life = 4 hours and if we want to calculate the amount after 9 hours (that's 2.25 half-lives) then:
ending amount = Bgng Amount / 2^n
ending amount = 50 / (2^(9/4))
ending amount = 50 / 2^2.25
ending amount = 50 / 4.75682846
ending amount = 10.5112 grams
Looking at the graph, we see that's about right.

Answer 2

a)
`y = 50(0.85)^x`

Explanation:

Let the function that shows the given situation is,

`y=ab^x`

Where a and b are any unknown numbers,

By the given diagram,

When x = 1, y = 42.5,

`\implies 42.5 = ab^1`

`\implies ab = 42.5` ------(1)

Again, when x = 4, y = 26,

`\implies 26 = ab^4`

`\implies 26 = ab(b^3)`

`\implies 26 = 42.5(b^3)` ( By equation (1) )

`\implies 0.611764706 = b^3`

`\implies 0.84890965425 = b`

Again, By equation (1),

a = 50.0642203646

Hence, the equation that shows the given graph,

`y = 50.0642203646(0.84890965425)^x`

`\implies y = 50(0.85)^x`

⇒ Option a is correct.