Sure! Let's start by defining the heights of the plants as functions of time, where [tex]\( t \)[/tex] represents the number of months.
Plant A:
- Initial height = 6 inches
- Growth rate = 3 inches per month
So, the height of Plant A after [tex]\( t \)[/tex] months, [tex]\( H_A(t) \)[/tex], is given by:
[tex]\[ H_A(t) = 6 + 3t \][/tex]
Plant B:
- Initial height = 4 inches
- Growth rate = 5 inches per month
So, the height of Plant B after [tex]\( t \)[/tex] months, [tex]\( H_B(t) \)[/tex], is given by:
[tex]\[ H_B(t) = 4 + 5t \][/tex]
We need to determine when Plant B will be taller than Plant A. To do this, we set up an inequality where the height of Plant B is greater than the height of Plant A:
[tex]\[ H_B(t) > H_A(t) \][/tex]
Substitute the expressions for [tex]\( H_A(t) \)[/tex] and [tex]\( H_B(t) \)[/tex]:
[tex]\[ 4 + 5t > 6 + 3t \][/tex]
Now, let's solve this inequality step-by-step:
1. Subtract [tex]\( 3t \)[/tex] from both sides to isolate the terms involving [tex]\( t \)[/tex] on one side:
[tex]\[ 4 + 5t - 3t > 6 + 3t - 3t \][/tex]
[tex]\[ 4 + 2t > 6 \][/tex]
2. Simplify the inequality:
[tex]\[ 2t + 4 > 6 \][/tex]
3. Subtract 4 from both sides to further isolate the term involving [tex]\( t \)[/tex]:
[tex]\[ 2t + 4 - 4 > 6 - 4 \][/tex]
[tex]\[ 2t > 2 \][/tex]
4. Divide both sides by 2 to solve for [tex]\( t \)[/tex]:
[tex]\[ \frac{2t}{2} > \frac{2}{2} \][/tex]
[tex]\[ t > 1 \][/tex]
Therefore, Plant B will be taller than Plant A when the number of months [tex]\( t \)[/tex] is greater than 1. So, the inequality that shows when Plant B will be taller than Plant A is:
[tex]\[ t > 1 \][/tex]
This means that after more than 1 month, Plant B will be taller than Plant A.
