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
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
1 answer