Final answer:
After solving the growth rate inequality between the oak tree and mesquite tree, it is determined that the oak tree will exceed the height of the mesquite tree after 1 year.
Step-by-step explanation:
We need to calculate when the oak tree's height will exceed the mesquite tree's height. Let's denote the initial height of the oak tree as O, the growth rate of the oak tree as G_o, the initial height of the mesquite tree as M, and the growth rate of the mesquite tree as G_m. We have:
- O = 4.7 feet
- G_o = 3.5 feet/year
- M = 5.2 feet
- G_m = 2.5 feet/year
Each year, the oak tree will be O + G_o × n feet tall, and the mesquite tree will be M + G_m × n feet tall, where n is the number of years. We want to find the smallest n such that O + G_o × n > M + G_m × n. Substituting the given values and simplifying, we get the following inequality:
4.7 + 3.5n > 5.2 + 2.5n
To find n, we solve for n in the inequality:
3.5n - 2.5n > 5.2 - 4.7
n > 0.5/1
n > 0.5
Since n must be a whole number of years, the smallest n that satisfies the inequality is 1. Therefore, after 1 year, the oak tree will exceed the height of the mesquite tree.