35fish/month means that after t months you get an extra 35t fish, right?
so, which choice reflects that?
a)R(t)=7000+35t-2000sin((pi/6)t)
b)R(t)=7000-2035sin((pi/6)t)
c)R(t)=7000-2000sin((pi/6)(t+35))
d)R(t)=7035-2000sin((pi/6)t)
so, which choice reflects that?
The original function is P(t) = 7000 - 2000sin((Ï€/6)t)
By adding the rate of increase to this, we get the new function R(t):
R(t) = 7000 - 2000sin((Ï€/6)t) + 35t
Therefore, the correct answer is option a) R(t) = 7000 + 35t - 2000sin((Ï€/6)t)
Therefore, the correct function would be:
R(t) = 7000 - 2000sin((pi/6)t) + 35t
Looking at the options provided:
a) R(t) = 7000 + 35t - 2000sin((pi/6)t) - This option matches our derived function, so it could be the correct answer.
b) R(t) = 7000 - 2035sin((pi/6)t) - This option does not add the rate of increase, 35t, to the original function. Hence, it is not the correct answer.
c) R(t) = 7000 - 2000sin((pi/6)(t+35)) - This option tries to incorporate the rate of increase, but it adds it as a factor inside the sine function. However, the rate of increase should be added separately to the function, not inside the sine function. Hence, it is not the correct answer.
d) R(t) = 7035 - 2000sin((pi/6)t) - This option includes the desired rate of increase, 35t, but it incorrectly changes the starting value of the fish population to 7035 instead of 7000. Therefore, it is not the correct answer.
Based on the analysis, option a) R(t) = 7000 + 35t - 2000sin((pi/6)t) is the correct function that could now be used to model the number of fish in the pond.