In 1994 the moose population in a park was measured to be 6400. By 1996, the population was measured again to be 6800. If the population continues to change linearly:
Find an equation for the moose population, y,in terms of x, the years since 1994.
The two-point form of a straight line passing through the points ( x1, y1 ) and ( x2 , y2 ) is given by:
y − y1 = ( y2 − y1 ) * ( x − x 1 ) / ( x2 − x1 )
Ini this case:
x1 = 1994 , x2 = 1996
y1 = 6400 , y2 = 6800
So:
y − y1 = ( y2 − y1 ) * ( x − x 1 ) / ( x2 − x1 )
y − 6400 = ( 6800 − 6400 ) * ( x − 1994 ) / ( 1996 − 1994 )
y − 6400 = 400 * ( x − 1994 ) / 2
y − 6400 = 200 * ( x − 1994 )
y − 6400 = 200 * x − 200 * 1994
y − 6400 = 200 x − 398800 Add 6400 to both sides
y − 6400 + 6400 = 200 x − 398800 + 6400
y = 200 x - 392400
Proof:
x = 1994
y = 200 x - 392400 = 200 * 1994 - 392400 = 398800 - 392400 = 6400
x = 1996
y = 200 x - 392400 = 200 * 1996 - 392400 = 399200 - 392400 = 6800
thank you
Now it is asking what will be the prediction for 2006?
To find the equation for the moose population in terms of the years since 1994, we can use the concept of linear equations.
First, let's define the initial year (1994) as the x-coordinate, and the corresponding population (6400) as the y-coordinate. We can represent this point as (x1, y1) = (0, 6400).
Next, we'll define another year (1996) as the second x-coordinate and its corresponding population (6800) as the second y-coordinate. This point can be represented as (x2, y2) = (2, 6800).
Now, we need to find the equation of the line passing through these two points. The equation for a line in slope-intercept form is given by y = mx + b, where m is the slope and b is the y-intercept.
To find the slope (m), we can use the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the given values:
m = (6800 - 6400) / (2 - 0) = 400 / 2 = 200
Now, we have the slope (m = 200).
Next, we need to find the y-intercept (b). We can use the equation of a line and substitute the values of one of the points:
6400 = 200 * 0 + b
6400 = b
Therefore, the y-intercept (b) is 6400.
Finally, we have both the slope (m = 200) and y-intercept (b = 6400), so the equation for the moose population in terms of the years since 1994 (x) is:
y = 200x + 6400