To determine how many days it takes for the sunfish to gain [tex]\(\frac{2}{7}\)[/tex] of its mass, we start by setting up the relationship given by the function:
[tex]\[ M(t) = 4 \cdot \left( \frac{81}{49} \right)^t \][/tex]
The mass increases by a factor of [tex]\(\frac{81}{49}\)[/tex] each day, so we want to find the number of days, [tex]\(t\)[/tex], where the mass increases by [tex]\(\frac{2}{7}\)[/tex].
First, we need to determine the new mass that corresponds to an increase of [tex]\(\frac{2}{7}\)[/tex]. An increase of [tex]\(\frac{2}{7}\)[/tex] of the current mass means the mass will be:
[tex]\[ M_{\text{new}} = M(t) \times \left(1 + \frac{2}{7}\right) = M(t) \times \frac{9}{7} \][/tex]
Next, we set up the equation for the increase factor:
[tex]\[ \left( \frac{81}{49} \right)^t = \frac{9}{7} \][/tex]
To solve for [tex]\(t\)[/tex], we use logarithms. Taking the natural logarithm on both sides,
[tex]\[ \ln \left( \left( \frac{81}{49} \right)^t \right) = \ln \left( \frac{9}{7} \right) \][/tex]
Using the property of logarithms that [tex]\(\ln(a^b) = b \ln(a)\)[/tex],
[tex]\[ t \cdot \ln \left( \frac{81}{49} \right) = \ln \left( \frac{9}{7} \right) \][/tex]
Solving for [tex]\(t\)[/tex],
[tex]\[ t = \frac{\ln \left( \frac{9}{7} \right)}{\ln \left( \frac{81}{49} \right)} \][/tex]
Calculating this, we get:
[tex]\[ t \approx 0.5 \][/tex]
Therefore, rounding to two decimal places,
The sunfish gains [tex]\(\frac{2}{7}\)[/tex] of its mass every [tex]\(0.50\)[/tex] days.
