Question

In 2000 the total amount of gamma ray bursts was recorded at 6.4 million for a city. In 2005, the same survey was made and the total amount of gamma ray bursts was 7.3 million. If the Earth can only withstand 1 billion gamma ray bursts, in what year will this become a problem? (Find the year in which the gamma ray bursts is 1 billion.)

Answer 1

In 2000 the total amount of gamma ray bursts was recorded at 6.4 million for a city.

We assume at 2000, t=0.

When t=0 then gamma ray bursts y= 6.4 million. that is (0,6.4)

In 2005, the same survey was made and the total amount of gamma ray bursts was 7.3 million

In 2005 , t= 5

When t=5 then gamma ray bursts y= 7.3 million. that is (5, 7.3)

Frame an equation using (0,6.4) and (5, 7.3)

Use equation y = a(b)^t and solve for a and b

(0,6.4) =>
`6.4=a(b)^0` so a= 6.4

(5, 7.3) => 7.3=6.4(b)^5

`(7.3)/(6.4) = b^5`

Now take fifth root on both sides

b= 1.0266

So the equation becomes
`y= 6.4(1.0266)^t`

Now we need to find out t when gamma ray bursts 1 billion

1 billion = 1000 millions

So we replace y by 1000 and we solve for t

`y= 6.4(1.0266)^t`

`1000= 6.4(1.0266)^t`

Divide by 6.4 and take ln on both sides

`ln((1000)/(6.4) )= ln(1.0266)^t`

We move the exponent 't' before ln

`ln((1000)/(6.4) )=t * ln(1.0266)`

Now divide both sides by ln(1.0266)

`(ln((1000)/(6.4)))/(ln(1.0266)) =t`

t = 192.419

It will take around 192 years

2000+ 192= 2192

In 2192, the gamma ray bursts is 1 billion