Question
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.
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
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
Answers
GPT-4o mini
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.
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.