Plant A is 4 cm tall and grows at a constant rate of 2 cm per month. Plant B is 8 cm tall and grows at a constant rate of 1 cm per month. The relationship between time in months, t, and height in cm, h, can be represented by the equation h=4+2t for Plant A and h=8+t for Plant B.

Question

The point of intersection is highlighted on the graph. What do the coordinates of the point of intersection represent in this context?
t
h
a
b
(4, 12)
0
1
2
3
4
5
6
7
8
9
10
0
2
4
6
8
10
12
14
16
18
20
22
24
Time in months
Height in cm
After
months, both plants are
cm

Answer 1

The point of intersection of the two lines representing the growth of Plant A and Plant B indicates the time at which both plants are of equal height.

From the equations given:

• Plant A: \( h = 4 + 2t \)
• Plant B: \( h = 8 + t \)

To find the point where they intersect, set the two equations equal to each other:

\[ 4 + 2t = 8 + t \]

Now, solve for \( t \):

1. Subtract \( t \) from both sides: \[ 4 + 2t - t = 8 \] \[ 4 + t = 8 \]

2. Subtract 4 from both sides: \[ t = 4 \]

Now, substitute \( t = 4 \) into either equation to find the height at this time. Using Plant A's equation: \[ h = 4 + 2(4) = 4 + 8 = 12 \]

So, the coordinates of the point of intersection are \( (4, 12) \).

In this context, \( t = 4 \) represents the time in months, and \( h = 12 \) represents the height in cm. Therefore:

After 4 months, both plants are 12 cm tall.