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 need to find the equation of the parabola that fits the given points.
We can use the general form of a quadratic function: P(x) = ax^2 + bx + c.
Plugging in the given points, we have:
When x = 0, P(x) = 350.
When x = 1, P(x) = 353.
When x = 2, P(x) = 382.
When x = 3, P(x) = 437.
When x = 4, P(x) = 518.
When x = 5, P(x) = 625.
When x = 6, P(x) = 758.
We can now create a system of equations using these data points:
350 = a(0)^2 + b(0) + c
353 = a(1)^2 + b(1) + c
382 = a(2)^2 + b(2) + c
437 = a(3)^2 + b(3) + c
518 = a(4)^2 + b(4) + c
625 = a(5)^2 + b(5) + c
758 = a(6)^2 + b(6) + c
Simplifying these equations, we get:
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
From the first equation, c = 350.
Substituting c into the second equation, we get:
353 = a + b + 350
3 = a + b
Now, substituting c = 350 into the other equations and simplifying, we get:
382 = 4a + 2b + 350
437 = 9a + 3b + 350
518 = 16a + 4b + 350
625 = 25a + 5b + 350
758 = 36a + 6b + 350
Subtracting 350 from each equation, we have:
32 = 4a + 2b
87 = 9a + 3b
168 = 16a + 4b
275 = 25a + 5b
408 = 36a + 6b
Now, we can solve this system of equations. One approach is to use the method of substitution or elimination. Another approach is to use a calculator or matrix manipulation software. Solving this system, we find that a = 13 and b = -10.
Therefore, the equation that models the data is:
P(x) = 13x^2 - 10x + 350.
Finally, to estimate the number of fish at the lake on week 8, we substitute x = 8 into the equation:
P(8) = 13(8)^2 - 10(8) + 350
P(8) = 13(64) - 80 + 350
P(8) = 832 - 80 + 350
P(8) = 1102.
So, 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.