Answer:
1. P = 420t +6020; in 2009, P = 15,260
2. h = -4/3t +140/3
Step-by-step explanation:
1. There are several different forms of the equation of a line that are useful to keep handy, either written down or memorized. For the moose problem, you are given two times and two populations and asked to write a linear equation relating them. <em>The 2-point form of the equation of a line is useful for this</em>.
That equation looks like ...
y = (y2 -y1)/(x2 -x1)(x -x1) +y1
You are given the points ...
(year, population) = (1991, 7700) and (1996, 9800)
You are asked to write the equation for population (P) in terms of "t" where "t" is years since 1987. Then the points are ...
(t, P) = (4, 7700) and (9, 9800)
Using these points in place of (x1, y1) and (x2, y2) in the above equation, we have ...
P = (9800 -7700)/(9 -4)(t -4) +7700
This is "an equation" for the moose population, but it is more useful if we simplify it a bit.
P = 2100/5(t -4) +7700
P = 420t + 6020 . . . . . . . . parentheses eliminated, like terms combined
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In 2009, t=22 years after 1987, so the population is predicted to be ...
P = 420·22 +6020 = 15,260
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2. Another useful form of the equation for a line is the <em>point-slope form</em>. That looks like ...
y -b = m(x -a) . . . . . . for slope m and point (a, b)
Parallel lines have the same slope, so the slope of the line you want is the slope of the line you have. That is the t-coefficient, -4/3. The expression h(20)=20 means that one of the points on the line is (t, h(t)) = (20, 20). So our point-slope equation for the line h(t) is ...
h -20 = -4/3(t -20)
Adding 20 and simplifying this, we get ...
h = -4/3t +140/3
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<em>Comment on the use of different variables</em>
When we're learning about lines and their equations, we often deal mainly with x and y and the x-y coordinate plane. You should realize that "x" is a sort of generic name for "any single independent variable." Likewise, "y" is a generic name for "any single dependent variable."
When we describe a function such as P(t) or h(t), we're saying the independent variable is named t (not x) and the dependent variable is named P or h (not y). In the above forms of linear equations, the generic variables are used. When you go to apply those forms to a specific problem, you need to use the independent and dependent variables of the problem in place of x and y.
