Question

Plant A starts at a height 25 cm tall and grows at a rate of 8 cm per month. At the same time, Plant B starts at a height of 10 cm and grows at a rate of 13 cm per month. Use the variable t to represent the number of months. If the plants continue to grow at this rate, after how many months will the plants be the same height? What height will they be at that time?
Write an expression to represent Plant A’s height after t months, where t represents the number of months.
Plant A

Plant B
Great job!
Now, write an expression to represent Plant B’s height after t months, where t represents the number of months.
Plant A

25 + 8t
Plant B

Good work!
You can set the expressions for Plant A’s height and Plant B’s height equal to one another to write an equation that represents Plant A and Plant B having an equal height. Do that now.
Plant A's height after t months: 25+8t
Plant B's height after t months: 10+13t

25+8t
=

10+13t
Plant A's height
Plant B's height
Excellent!
Use the equation to solve for t to find how many months it will take for Plant A and Plant B to be the same height.
Plant A's height after t months: 25+8t
Plant B's height after t months: 10+13t
25+8t
=
10+13t

t
=
Great job!
What does your answer of 3 mean in this problem?
Plant A's height after t months: 25+8t
Plant B's height after t months: 10+13t
25 + 8t = 10 + 13t
t = 3

Plant A and Plant B will be 3 cm tall at the same time.

Plant A and Plant B will be the same height after 3 months.

Answer 1

To find how many months it will take for Plant A and Plant B to be the same height, we first set up the equations based on their growth:

For Plant A:
- Initial height: 25 cm
- Growth rate: 8 cm per month
- Height after \( t \) months:
\[
\text{Height of Plant A} = 25 + 8t
\]

For Plant B:
- Initial height: 10 cm
- Growth rate: 13 cm per month
- Height after \( t \) months:
\[
\text{Height of Plant B} = 10 + 13t
\]

Setting these two expressions equal to each other to find when they will be the same height:
\[
25 + 8t = 10 + 13t
\]

Now, let's solve for \( t \):

1. Subtract \( 8t \) from both sides:
\[
25 = 10 + 5t
\]

2. Subtract 10 from both sides:
\[
15 = 5t
\]

3. Divide both sides by 5:
\[
t = 3
\]

This means that after **3 months**, both plants will be the same height.

Next, we can find the height of both plants at that time by substituting \( t = 3 \) back into either equation:

Using Plant A's height equation:
\[
\text{Height of Plant A} = 25 + 8(3) = 25 + 24 = 49 \text{ cm}
\]

Using Plant B's height equation:
\[
\text{Height of Plant B} = 10 + 13(3) = 10 + 39 = 49 \text{ cm}
\]

Both plants will be **49 cm tall** after **3 months**.

So, the answers are:
- After **3 months**, Plant A and Plant B will be the same height.
- Their height at that time will be **49 cm**.