The number of waterfowl on 8 week would be \(\boxed{2555}\).
Further explanation:
The general form of the quadratic equation is given by,
\(\boxed{a{x^2} + bx + c}\)
Here, a is the coefficient of \({x^2}\), b is the coefficient of \(x\) and \(c\) is the constant term.
Given:
The population of the lake in 0, 1, 2, 3, 4, 5 and 6 week is 635, 644, 719, 860, 1067, 1340 and 1769 respectively.
Calculation:
Consider \(x\) denote the number of weeks.
Quadratic function that models the data as the function of \(x\) can be expressed as,
\(\boxed{N=a{x^2}+bx +c}\)
Here, \(N\) denotes the population in \({n^{th}}\) week.
The population of lake in the beginning is 635. Therefore, value of \(x\) is zero and value of \(N\) is 635.
Substitute \(0\) for \(x\) and \(635\) for \(N\) in quadratic equation \(N = a{x^2} + bx + c\).
\(\begin{aligned} 635&=a{\left(0\right)^2}+b\left( 0 \right)+c\\ 635&=0+0+c\\ 635&=c\\\end{aligned}\)
The population of lake after one week is \(644\). Therefore, value of \(x\) is one and value of \(N\) is \(644\).
Substitute \(1\) for \(x\) and \(644\) for \(N\) in quadratic equation \(N = a{x^2} + bx + c\).
\(\begin{aligned} 644 &= a{\left( 1 \right)^2} + b\left( 1 \right) + c \\ 644 &= a + b + c \\ 644 &= a + b + 635 \\ 644 - 635 &= a + b \\ 9 &= a + b \\\end{aligned}\)
The population of lake after two week is \(719\). Therefore, value of \(x\) is two and value of \(N\) is \(719\).
Substitute 2 for \(x\) and 719 for \(N\) in quadratic equation \(N = a{x^2} + bx + c\).
\(\begin{aligned} 719 &= a{\left( 2 \right)^2} + b\left( 2 \right) + c \\ 719 &= 4a + 2b + 635 \\ 719 - 635 &= 4a + 2b \\ 84 &= 2\left( {2a + b} \right) \\ \frac{{84}}{2} &= \left( {2a + b} \right) \\ 42 &= \left( {2a + b} \right) \\ \end{aligned}\)
Solve the equation \(a + b = 9\) to obtain the value \(a\).
\(a = 9 - b\)
Substitute \(9 - b\) for in equation \(42 = 2a + b\).
\(\begin{aligned} 42 &= 2\left( {9 - b} \right) + b \\42 &= 18 - 2b + b \\ 42 - 18 &=- b \\24 &=- b\\- 24 &= b \\ \end{aligned}\)
Substitute \(-24\) for \(b\) in equation \(a = 9 - b\) to obtain the value of a.
\(\begin{aligned}a&= 9-\left( {-24} \right) \\&= 9+24\\&= 33 \\\end{aligned}\)
To obtain the quadratic function substitute 33 for a, -24 for b and 635 for c in equation \(N = a{x^2} + bx + c\).
\(N = 33{x^2} - 24x + 635\)
Substitute 8 for \(x\) in above equation to obtain the number of waterfowl at the lake after week \(8\).
\(\begin{aligned} N &= 33{\left( 8 \right)^2} - 24\left( 8 \right) + 635 \\ &=33\left( {64} \right) - 192 + 635 \\ &= 2112 - 192 + 635 \\ &= 2555 \\ \end{aligned}\)
The number of waterfowl on \(8\) week would be \(\boxed{2555}\).
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Answer details:
Grade: Middle School
Subject: Mathematics
Chapter: Quadratic equation
Keywords: quadratic equation, polynomial, square root, real numbers, coefficients, constant term, general form of quadratic equation, waterfowl, lake, population, migrating, a biologist, model, estimate, week, recounted, function.
