equation of the line

Question

If a tree is 12 feet tall after 2 years and 18 feet tall after 5 years, find the equation of the line describing its growth, where y is the height and x is the number of years. The equation is?

Answer 1

Imagine that the given information represents points in an x-y coordinate plane.

The points are: (2, 12) and ( 5, 18). The height of the tree depends on the number of years it has been growing.

Now find the slope of that line through those two points:

m = (18 - 12)/(5 -2) = 6/3 = 2 , that is for every year the tree grows its height goes up by 2.

Now use point-slope or slope-intercept to write the equation.

Here is slope-intercept using (2, 12) as the point.

y - 12 = 2(x - 2)

y = 2x - 4 +12

y = 2x + 8

Here is using slope-intercept:

12 = 2(2) + b

b = 8

So y = 2x + 8

Check the points:

When x = 2

y = 2(2) + 8 = 12

When x = 5

y = 2(5) + 8 = 18

Now you can use the equation to find the height after different number of years.

After 8 years height is:

2(8) + 8 = 24

Of course that cannot go on forever!