To find a quadratic function that models the data, we can use the given population values to find the coefficients of the quadratic function.
Using the equation: P(x) = ax^2 + bx + c
We can substitute the x and P(x) values to get a system of equations:
350 = a(0)^2 + b(0) + c -> c = 350
353 = a(1)^2 + b(1) + c -> a + b + c = 353 - equation 1
382 = a(2)^2 + b(2) + c -> 4a + 2b + c = 382 - equation 2
437 = a(3)^2 + b(3) + c -> 9a + 3b + c = 437 - equation 3
Subtracting equation 1 from equation 2 and equation 3, we get two more equations:
4a + 2b = 29 - equation 4
9a + 3b = 87 - equation 5
Multiplying equation 4 by 3 and equation 5 by 2 and subtracting them, we eliminate b:
12a + 6b = 87
18a + 6b = 58
-----------------
-6a = -29
a = 29/6
Substituting a = 29/6 into equation 4, we can find b:
4(29/6) + 2b = 29
116/6 + 2b = 29
2b = 29 - 116/6
2b = 174/6 - 116/6
2b = 58/6
b = 58/12
b = 29/6
Therefore, the quadratic function that models the data is:
P(x) = (29/6)x^2 + (29/6)x + 350
To estimate the number of fish at the lake on week 8, we can substitute x = 8 into the quadratic function:
P(8) = (29/6)(8)^2 + (29/6)(8) + 350
P(8) = (29/6)(64) + (29/6)(8) + 350
P(8) = 371 + 464/6 + 350
P(8) = 371 + 77 + 350
P(8) = 798 + 77
P(8) = 875
Therefore, the estimated number of fish at the lake on week 8 is 875 fish.
Thus, 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
1 answer