Final answer:
It will take 12 decades for the Galveston and Florida Shore Lines to have the same amount of beach, by setting up and solving a linear equation for each shoreline's beach erosion.
Step-by-step explanation:
To find out after how many decades the Galveston Shore Line and the Florida Shore Line will have the same amount of beach, we can set up a linear equation for each shore line and solve for the time when they are equal. Let's call the number of decades 'd'.
Galveston Shore Line equation: G(d) = 20 - 2d, where 20 is the initial number of miles of beach and 2d represents the loss of 2 miles of beach per decade.
Florida Shore Line equation: F(d) = 14 - 1.5d, where 14 is the initial number of miles of beach and 1.5d represents the loss of 1.5 miles of beach per decade.
To find when G(d) equals F(d), we set them equal to each other:
20 - 2d = 14 - 1.5d
Solving for 'd':
20 - 14 = 2d - 1.5d6 = 0.5dd = 12
It will take 12 decades for the two shore lines to have the same amount of beach.