Answer:
Half (1/2)
Step-by-step explanation:
The relation between the Mass, M(t), of the ocean sunfish and time t, in days, since it is born is given as:
[tex]M(t)=3(\frac{81}{49})^{t}[/tex]
This expression shows that the mass of sunfish increases by a factor of [tex]\frac{81}{49}[/tex] after each day. For example one day 1, the mass of sunfish would be:
[tex]M(1)=3(\frac{81}{49} )[/tex]
And on day 2:
[tex]M(2)=3(\frac{81}{49} )^{2}[/tex]
We can re-write the given equation as:
[tex]M(t)=3(\frac{9^{2}}{7^{2}} )^{t}\\\\ M(t)=3((\frac{9}{7})^{2})^{t}\\\\ M(t)=3(\frac{9}{7})^{2t}\\[/tex]
The exponent has changed from t to 2t. The term "2t" means 2 times in a unit time t. So this means, the mass of sunfish increases by a factor of 9/7 every 1/2(half) day.
On Day 1, the mass would be:
[tex]M(t)=3(\frac{9}{7})^{2}[/tex]
After 1.5 day, the mass would be:
[tex]M(t)=3(\frac{9}{7})^{1.5 \times 2}\\\\ M(t)=3(\frac{9}{7} )^{3}[/tex]
This example shows that the mass of sunfish increases by a factor of 9/7 after every half of the day.
