Respuesta :
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) => [tex]6.4=a(b)^0[/tex] so a= 6.4
(5, 7.3) => 7.3=6.4(b)^5
[tex]\frac{7.3}{6.4} = b^5[/tex]
Now take fifth root on both sides
b= 1.0266
So the equation becomes [tex]y= 6.4(1.0266)^t[/tex]
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
[tex]y= 6.4(1.0266)^t[/tex]
[tex]1000= 6.4(1.0266)^t[/tex]
Divide by 6.4 and take ln on both sides
[tex]ln(\frac{1000}{6.4} )= ln(1.0266)^t[/tex]
We move the exponent 't' before ln
[tex]ln(\frac{1000}{6.4} )=t * ln(1.0266)[/tex]
Now divide both sides by ln(1.0266)
[tex]\frac{ln(\frac{1000}{6.4})}{ln(1.0266)} =t[/tex]
t = 192.419
It will take around 192 years
2000+ 192= 2192
In 2192, the gamma ray bursts is 1 billion
