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
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
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-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.
To find a quadratic function that models the data, we need to use the given population values at each week.
Week 0 1 2 3 4 5 6
Population 350 353 382 437 518 625 758
To determine the quadratic function, we'll use the general form of a quadratic function, P(x) = ax^2 + bx + c, where P(x) represents the population at week x.
Using the given data, we can now create a system of equations:
Week 0: P(0) = 350 = c
Week 1: P(1) = a(1^2) + b(1) + c = a + b + 350 = 353
Week 2: P(2) = a(2^2) + b(2) + c = 4a + 2b + 350 = 382
Week 3: P(3) = a(3^2) + b(3) + c = 9a + 3b + 350 = 437
Week 4: P(4) = a(4^2) + b(4) + c = 16a + 4b + 350 = 518
Week 5: P(5) = a(5^2) + b(5) + c = 25a + 5b + 350 = 625
Week 6: P(6) = a(6^2) + b(6) + c = 36a + 6b + 350 = 758
We have six equations with three variables (a, b, and c). To solve this system of equations, we can use matrix methods or substitution. Let's use substitution:
From the first equation, we know that c = 350.
Substituting c = 350 into the second equation, we get:
a + b + 350 = 353
a + b = 353 - 350
a + b = 3
Substituting c = 350 into the third equation, we get:
4a + 2b + 350 = 382
4a + 2b = 382 - 350
4a + 2b = 32
2a + b = 16
b = 16 - 2a
Substituting c = 350 into the fourth equation, we get:
9a + 3b + 350 = 437
9a + 3b = 437 - 350
9a + 3b = 87
3a + b = 29
b = 29 - 3a
Since b is equal to both 16 - 2a and 29 - 3a, we can set these two expressions equal to each other:
16 - 2a = 29 - 3a
a = 13
Substituting a = 13 into b = 16 - 2a, we get:
b = 16 - 2(13)
b = 16 - 26
b = -10
Now we know a = 13, b = -10, and c = 350, so 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 need to evaluate P(x) when x = 8:
P(8) = 13(8^2) - 10(8) + 350
P(8) = 13(64) - 10(8) + 350
P(8) = 832 - 80 + 350
P(8) = 1,102
Therefore, the estimated number of fish at the lake on week 8 is 1,102 fish.
The correct answer is: P(x) = 13x^2 - 10x + 350; 1,102 fish.
To find a quadratic function that models the data, we need to first create a table of values, and then use these values to solve a system of equations.
Week (x) | Population (y)
0 | 350
1 | 353
2 | 382
3 | 437
4 | 518
5 | 625
6 | 758
Now, let's use these values to set up a system of equations. We'll use the general form of a quadratic function, which is P(x) = ax^2 + bx + c.
Using the first point (0, 350):
350 = a(0)^2 + b(0) + c
350 = c
So, we have c = 350.
Using the second point (1, 353):
353 = a(1)^2 + b(1) + 350
353 = a + b + 350
a + b = 3
Using the third point (2, 382):
382 = a(2)^2 + b(2) + 350
382 = 4a + 2b + 350
4a + 2b = 32
2a + b = 16
Now, we have a system of equations:
a + b = 3
2a + b = 16
Solving this system of equations, we find a = 13 and b = -10.
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 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
So, the model estimates that there would be 1,102 fish at the lake on week 8. Therefore, the correct response is:
P(x) = 13x^2 - 10x + 350; 1,102 fish.