Brittany buys two trees for her backyard, one Oak tree and one Pine tree. She plants them appropriately and measures their initial height. Every year Brittany measures the height of the two trees and eventually realizes that the two trees must have had the same height at some point! Construct two linear equations in the form y=mx+b from the table and determine when the trees were the same height.
Let's assume that the initial height of the Oak tree is 1 meter and the Pine tree is 0.5 meters. Let x represent the number of years after planting the trees and y be the height of the trees in meters.
For the Oak tree:
y = 1 + mx
For the Pine tree:
y = 0.5 + nx
After 5 years, the Oak tree is 2 meters tall and the Pine tree is 2.5 meters tall. Therefore, we can construct the following equations based on the given information:
2 = 1 + 5m
2.5 = 0.5 + 5n
Solving these equations, we get:
m = 0.2
n = 0.5
To determine when the trees were the same height, we set the two equations equal to each other:
1 + 0.2x = 0.5 + 0.5x
Solving for x, we get:
0.3x = 0.5 - 1
0.3x = -0.5
x = -0.5 / 0.3
x = -1.67
Therefore, the trees were the same height after approximately 1.67 years.