Answer:
y = -2(x - 20)² + 6000
Step-by-step explanation:
Given: The maximum population, 6,000 people, occurred 20 years after record keeping began. Five years later, the population was 5,950
To Find: If x represents the number of years after record keeping began, which equation represents the given scenario?
Solution:
Vertex = (20,6000) and the parabola opens downward.
So, Equation is:
y = A (x - 20)² + 6000 --1
Now, we need to find A.
It is also given that Five years later, the population was 5,950
This means in 25th year the population was 5950.
So, point (25,5950) is on the parabola. Now substitute the given point in the equation to get value of A
5950 = A (25 - 20)² + 6000
-50 = A (5)²
-50 = 25 A
A = -2
Now substitute the value of A in 1
Thus Equation becomes: y = -2(x - 20)² + 6000
Hence Option 3 is correct .
y = -2(x - 20)² + 6000 represent the situation.
The equation represents the given scenario is [tex]\rm y = -2 (x - 20)^2+ 6000[/tex].
Given
The population, y, of a small town can be represented using a quadratic model.
The maximum population, 6,000 people, occurred 20 years after record-keeping began.
Five years later, the population was 5,950.
The vertex of a parabola is the point at which the parabola passes through its axis of symmetry.
The vertex = (20,6000) and the parabola open downward.
Then,
The equation is;
[tex]\rm y = A (x - 20)^2+ 6000[/tex]
After 5 years later means = 20 years + 5 years = 25 years
The population was 5,950.
Substitute x = 25 and y = 5950 in the equation
[tex]\rm y = A (x - 20)^2+ 6000\\\\5950 = A (25-20)^2 + 6000\\\\5950 - 6000= 25A\\\\A = \dfrac{-50}{25}\\\\A = -2[/tex]
Substitute the value of A in the equation
[tex]\rm y = A (x - 20)^2+ 6000\\\\\rm y = -2 (x - 20)^2+ 6000[/tex]
Hence, the equation represents the given scenario is [tex]\rm y = -2 (x - 20)^2+ 6000[/tex].
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