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?

Question

Write an expression to represent Plant A’s height after t months, where t represents the number of months.
Plant A

Answer 1

To represent Plant A’s height after \( t \) months, we can use the following expression:

\[ \text{Height of Plant A} = 25 + 8t \]

Where:

• 25 cm is the starting height of Plant A.
• 8 cm is the growth rate per month.
• \( t \) represents the number of months.

Similarly, we can write an expression for Plant B’s height after \( t \) months:

\[ \text{Height of Plant B} = 10 + 13t \]

Where:

• 10 cm is the starting height of Plant B.
• 13 cm is the growth rate per month.

To find out after how many months the plants will be the same height, we need to set the two expressions equal to each other:

\[ 25 + 8t = 10 + 13t \]

Now, we will solve for \( t \):

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

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

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

Now that we have \( t = 3 \), we can find the height of the plants at this time by substituting \( t \) back into either height expression.

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

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

Thus, after 3 months, both Plant A and Plant B will be the same height of 49 cm.

In summary:

• They will be the same height after 3 months.
• At that time, their height will be 49 cm.