A biologist took a count of the number of fish in a particular lake, and recounted the lake’s population of fish on each of the next six weeks.
Week
0
1
2
3
4
5
6
Population
350
353
382
437
518
625
758
Find a quadratic function that models the data as a function of x, the number of weeks. Use the model to estimate the number of fish at the lake on week 8.
To find a quadratic function that models the data, we need to find the equation of a parabola that fits the given data points. Since the population of fish is increasing over the weeks, we can expect the parabola to open upwards.
We have the following data points:
Week 0, Population 350
Week 1, Population 353
Week 2, Population 382
Week 3, Population 437
Week 4, Population 518
Week 5, Population 625
Week 6, Population 758
We can use these data points to form a system of equations. Let's represent the week as x and the population as y.
Using the first data point, we have:
x = 0, y = 350
Using the second data point, we have:
x = 1, y = 353
Using the third data point, we have:
x = 2, y = 382
Using the fourth data point, we have:
x = 3, y = 437
Using the fifth data point, we have:
x = 4, y = 518
Using the sixth data point, we have:
x = 5, y = 625
Using the seventh data point, we have:
x = 6, y = 758
Let's substitute these values into the general form of a quadratic function: y = ax^2 + bx + c.
For the first point, we get:
350 = a(0)^2 + b(0) + c
350 = c
For the second point, we get:
353 = a(1)^2 + b(1) + c
353 = a + b + c
For the third point, we get:
382 = a(2)^2 + b(2) + c
382 = 4a + 2b + c
For the fourth point, we get:
437 = a(3)^2 + b(3) + c
437 = 9a + 3b + c
For the fifth point, we get:
518 = a(4)^2 + b(4) + c
518 = 16a + 4b + c
For the sixth point, we get:
625 = a(5)^2 + b(5) + c
625 = 25a + 5b + c
For the seventh point, we get:
758 = a(6)^2 + b(6) + c
758 = 36a + 6b + c
Now we have a system of equations:
350 = c
353 = a + b + c
382 = 4a + 2b + c
437 = 9a + 3b + c
518 = 16a + 4b + c
625 = 25a + 5b + c
758 = 36a + 6b + c
Substitute c = 350 into the other equations:
353 = a + b + 350
382 = 4a + 2b + 350
437 = 9a + 3b + 350
518 = 16a + 4b + 350
625 = 25a + 5b + 350
758 = 36a + 6b + 350
Simplify these equations:
3 = a + b
32 = 4a + 2b
87 = 9a + 3b
168 = 16a + 4b
275 = 25a + 5b
408 = 36a + 6b
Solve these equations for a and b. We can subtract 3 from each of the equations, then divide by the corresponding coefficients:
a = 0
b = 3
Now we have the values for a, b, and c in the quadratic function:
y = ax^2 + bx + c
y = 0x^2 + 3x + 350
y = 3x + 350
To estimate the number of fish in week 8, we can substitute x = 8 into the equation:
y = 3(8) + 350
y = 24 + 350
y = 374
Therefore, the estimated number of fish in week 8 is 374.