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.)

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)  => so a= 6.4

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

Now take fifth root on both sides

b= 1.0266

So the equation becomes

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

Divide by 6.4  and take ln on both sides

We move the exponent 't' before ln

Now divide both sides by ln(1.0266)

t = 192.419

It will take around 192 years

2000+ 192= 2192

 In 2192, the gamma ray bursts is 1 billion

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