Question

Ocean sunfish are well known for rapidly gaining a lot of weight on a diet based on jellyfish. The relationship between the elapsed time, t, in days, since an ocean sunfish is born, and it’s mass, M(t), in milligrams, is modeled by the following function:
M(t)=(1.34)^t/6+4

Complete the following sentence about the daily rate of change in the mass of the sunfish. Round your answer to two decimal places.

Every day, the mass of the sunfish is multiplied by a factor of _______.

Answer 1

Answer:

Every day, the mass of the sunfish is multiplied by a factor of [ln(1.34)/6].

Step-by-step explanation:

You have the following function:

$M(t)=(1.34)^{\frac{t}{6}+4}$

To know what is the factor that multiplies the mass of the sunfish each day, you derivative the function M(t):

$\frac{dM(t)}{dt}=(1.34)^{\frac{t}{6}+4}(\frac{1}{6})ln(1.34)\\\\\frac{dM(t)}{dt}=\frac{ln(1.34)}{6}[(1.43)^{\frac{t}{6}+4}]\\\\\frac{dM(t)}{dt}=\frac{ln(1.34)}{6}M(t)$  (1)

where you have used the following general derivative:

$g(t)=a^{f(t)}\\g'(t)=a^{f(t)}f'(t)ln(a)$

The derivative give you the increase in the mass per day (because t is days). By the expression (1) you can conclude that each day the mass increase a factor of [ln(1.34)/6].

Every day, the mass of the sunfish is multiplied by a factor of [ln(1.34)/6].