Gilberto plants two trees in his front yard. The apple tree is three feet tall and will grow 20 percent taller each year. The olive tree is two feet tall and will grow 30 percent taller each year. Create equations that model each tree’s height per year. How many years will it take for the trees to reach the same height?(1 point) Responses approximately 42 years approximately 42 years approximately 7 years approximately 7 years approximately 8 years approximately 8 years approximately 5 years
Let A(t) represent the height of the apple tree at time t in years, and let O(t) represent the height of the olive tree at time t in years.
For the apple tree:
A(t) = 3(1.20)^t
For the olive tree:
O(t) = 2(1.30)^t
We want to find the time t when A(t) = O(t):
3(1.20)^t = 2(1.30)^t
Divide by 2:
1.5(1.20)^t = (1.30)^t
Divide by 1.5:
(1.20)^t = (1.30)^t/1.5
Take the natural log of both sides:
ln(1.20)^t = ln((1.30)^t/1.5)
t*ln(1.20) = ln((1.30)^t/1.5)
t = ln((1.30)^t/1.5)/ln(1.20)
Approximating this value gives t ≈ 7 years. Therefore, it will take approximately 7 years for both trees to reach the same height.