francois is stokcing a new fish pond on his farm. The pond is filling slowly and the volume of water in it can support a certain number of fish,(variable-F). F=42m+250, where m is the number of months the pond has been filling.
the pond was stocked initially with 110 fish, a population that will grow at a rate of 8 percent per month. P=110(1.08) to the power of m. this is the equation that models the population of fish after m number of months.
when will p equal f?
what will happen after this point has been reached? explain your answer.
P = F when
42 m + 250 = 110*1.08^m
That equation can be solved by iteration. If m = 30,
42 m + 250 = 1510
110*1.08^m = 1106
If m = 35
F = 42 m + 250 = 1720
P = 110*1.08^m = 1626
If m = 36
42 m + 250 = 1762
110*1.08^m = 1756
m is slightly more than 36 months
At longer times, P will exceed F, and fish will start to die because the pond is too small to support the population.
To find out when P will equal F, we can set the two equations equal to each other and solve for m.
P = 110(1.08)^m
F = 42m + 250
Setting P equal to F:
110(1.08)^m = 42m + 250
To solve this equation, you can use algebraic methods or approximation methods such as graphing or using a numerical solver. Let's solve it using algebraic methods:
110(1.08)^m = 42m + 250
Divide both sides by 110:
(1.08)^m = (42m + 250) / 110
Take the natural logarithm of both sides to solve for m:
ln[(1.08)^m] = ln[(42m + 250) / 110]
Using the logarithm property ln(a^b) = b * ln(a):
m * ln(1.08) = ln[(42m + 250) / 110]
Divide both sides by ln(1.08):
m = ln[(42m + 250) / 110] / ln(1.08)
Now, you can plug this equation into a graphing calculator or a numerical solver to find the approximate value of m when P equals F.
Once this point is reached, it means that the population of fish (P) is equal to the volume of water in the pond can support (F). At this point, the pond is considered fully stocked, and no additional fish can be added without exceeding the capacity of the pond. It's important to maintain the population below this level to ensure the well-being and sustainability of the fish population and the overall ecosystem of the pond.