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.
(1 point)
Responses
P(x) = 13x2 – 10x + 350; 917 fish
P ( x ) = 13 x 2 – 10 x + 350; 917 fish
P(x) = 13x2 – 10x + 350; 1,102 fish
P ( x ) = 13 x 2 – 10 x + 350; 1,102 fish
P(x) = 18x2 + 10x + 300; 1,252 fish
P ( x ) = 18 x 2 + 10 x + 300; 1,252 fish
P(x) = 18x2 + 10x + 300; 1,532 fish
P ( x ) = 18 x 2 + 10 x + 300; 1,532 fish
To find a quadratic function that models the data, we can use the general form of a quadratic function: P(x) = ax^2 + bx + c.
Using the given data points, we have the following equations:
P(0) = 350
P(1) = 353
P(2) = 382
P(3) = 437
P(4) = 518
P(5) = 625
P(6) = 758
Substituting these values into the function, we get the following system of equations:
a(0)^2 + b(0) + c = 350 (Equation 1)
a(1)^2 + b(1) + c = 353 (Equation 2)
a(2)^2 + b(2) + c = 382 (Equation 3)
a(3)^2 + b(3) + c = 437 (Equation 4)
a(4)^2 + b(4) + c = 518 (Equation 5)
a(5)^2 + b(5) + c = 625 (Equation 6)
a(6)^2 + b(6) + c = 758 (Equation 7)
Simplifying these equations, we get:
c = 350 (Equation 8)
a + b + c = 353 (Equation 9)
4a + 2b + c = 382 (Equation 10)
9a + 3b + c = 437 (Equation 11)
16a + 4b + c = 518 (Equation 12)
25a + 5b + c = 625 (Equation 13)
36a + 6b + c = 758 (Equation 14)
Substituting Equation 8 into Equations 9-14, we can eliminate c:
a + b = 3 (Equation 15)
4a + 2b = 32 (Equation 16)
9a + 3b = 87 (Equation 17)
16a + 4b = 168 (Equation 18)
25a + 5b = 275 (Equation 19)
36a + 6b = 408 (Equation 20)
Solving this system of equations, we find that a = 13 and b = -10. Substituting these values back into Equation 8, we get c = 350.
Therefore, the quadratic function that models the data is:
P(x) = 13x^2 - 10x + 350
To estimate the number of fish in the lake on week 8, we substitute x = 8 into the function:
P(8) = 13(8)^2 - 10(8) + 350
P(8) = 13(64) - 80 + 350
P(8) = 832 - 80 + 350
P(8) = 1102
Therefore, the estimated number of fish in the lake on week 8 is 1,102.
The correct answer is: P(x) = 13x^2 - 10x + 350; 1,102 fish
To find a quadratic function that models the data, we can use the formula P(x) = ax^2 + bx + c, where P(x) represents the population of fish at week x.
Using the given data points (0, 350), (1, 353), and (2, 382), we can set up a system of equations:
350 = a(0)^2 + b(0) + c
353 = a(1)^2 + b(1) + c
382 = a(2)^2 + b(2) + c
Simplifying the equations, we get:
350 = c
a + b + c = 353
4a + 2b + c = 382
Using these equations, we can solve for a, b, and c:
a = 11
b = -10
c = 350
Therefore, the quadratic function that models the data is P(x) = 11x^2 - 10x + 350.
To estimate the number of fish at the lake on week 8, we can substitute x = 8 into the function:
P(8) = 11(8)^2 - 10(8) + 350
P(8) = 704 - 80 + 350
P(8) = 974
Therefore, the estimated number of fish at the lake on week 8 is 974 fish.
So, the correct answer is:
P(x) = 11x^2 - 10x + 350; 974 fish
To find a quadratic function that models the data, we need to identify the pattern and relationships between the weeks and the population of fish.
Looking at the data, we can see that the population of fish is increasing over time. This suggests a positive quadratic function.
To find the quadratic function, we can use the formula: P(x) = ax^2 + bx + c, where P(x) represents the population of fish at week x.
We can start by substituting the given data points into the formula to create a system of equations:
P(0) = a(0)^2 + b(0) + c = 350
P(1) = a(1)^2 + b(1) + c = 353
P(2) = a(2)^2 + b(2) + c = 382
P(3) = a(3)^2 + b(3) + c = 437
P(4) = a(4)^2 + b(4) + c = 518
P(5) = a(5)^2 + b(5) + c = 625
P(6) = a(6)^2 + b(6) + c = 758
Simplifying these equations, we get:
c = 350
a + b + c = 353
4a + 2b + c = 382
9a + 3b + c = 437
16a + 4b + c = 518
25a + 5b + c = 625
36a + 6b + c = 758
Now we can solve this system of equations. One way to do this is by using matrix methods or using a calculator that can solve systems of equations.
By solving the system of equations, we find that a = 13, b = -10, and c = 350.
Therefore, the quadratic function that models the data is:
P(x) = 13x^2 - 10x + 350.
To estimate the number of fish at the lake on week 8, we can substitute x = 8 into the quadratic function:
P(8) = 13(8)^2 - 10(8) + 350
P(8) = 13(64) - 80 + 350
P(8) = 832 - 80 + 350
P(8) = 1102.
Therefore, the estimated number of fish at the lake on week 8 is 1,102 fish.
So the correct answer is: P(x) = 13x^2 - 10x + 350; 1,102 fish.