In the past, the number of fish in a pond could be modelled by the function P(t)=7000-2000sin((pi/6)t) where t is the number of months after January. However, because of a lack of fishermen since the start of the new year, the number of fish in the pond has been increasing at a rate of 35 per month. If this pattern continues, which of these functions could now be used to model the number of fish in the pond?
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)
To determine the function that models the number of fish in the pond now, we need to add the constant rate of increase of 35 fish per month to the original function P(t).
The original function is P(t) = 7000 - 2000sin((π/6)t)
By adding the rate of increase to this, we get the new function R(t):
R(t) = 7000 - 2000sin((π/6)t) + 35t
Therefore, the correct answer is option a) R(t) = 7000 + 35t - 2000sin((π/6)t)
To find the new function that models the number of fish in the pond, we need to add the rate of increase in the number of fish to the original function P(t). We can do this by adding 35t to the function P(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.
35fish/month means that after t months you get an extra 35t fish, right?
so, which choice reflects that?