Students graphed the growth rate over several weeks of two plants in their classroom. The equations of both plants are given where [tex]x[/tex] represents the time in weeks and [tex]y[/tex] represents the heights of the plants in inches.

Plant A: [tex]y = 1.8x + 3.1[/tex]
Plant B: [tex]y = 2.3x + 1.9[/tex]

Approximately how many weeks will it take for both plants to reach the same height? Round your answer to the nearest tenth.

A. 1.9 weeks
B. 2.4 weeks
C. 3.1 weeks
D. 7.4 weeks



Answer :

To determine when both plants will reach the same height, we need to set the two equations equal to each other and solve for \( x \). The equations representing the plants' heights are:

[tex]\[ \text{Plant A: } y = 1.8x + 3.1 \][/tex]
[tex]\[ \text{Plant B: } y = 2.3x + 1.9 \][/tex]

We set the equations equal to each other to find the point where the heights are the same:

[tex]\[ 1.8x + 3.1 = 2.3x + 1.9 \][/tex]

Next, we solve for \( x \):

1. Subtract \( 1.8x \) from both sides:
[tex]\[ 3.1 = 2.3x - 1.8x + 1.9 \][/tex]
[tex]\[ 3.1 = 0.5x + 1.9 \][/tex]

2. Subtract \( 1.9 \) from both sides:
[tex]\[ 3.1 - 1.9 = 0.5x \][/tex]
[tex]\[ 1.2 = 0.5x \][/tex]

3. Divide both sides by 0.5:
[tex]\[ x = \frac{1.2}{0.5} \][/tex]
[tex]\[ x = 2.4 \][/tex]

Therefore, it will take approximately 2.4 weeks for both plants to reach the same height. Rounding this to the nearest tenth confirms our answer.

So, the correct option is:
[tex]\[ \boxed{2.4 \text{ weeks}} \][/tex]