Plant A starts at a height 75 cm tall and grows at a rate of 10 cm per month. At the same time, Plant B starts at a height of 55 cm and grows at a rate of 12 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

Plant B

Answer 1

To represent the heights of Plant A and Plant B after \( t \) months, we can write the following expressions:

1. Plant A's height after \( t \) months: \[ \text{Height of Plant A} = 75 + 10t \]

2. Plant B's height after \( t \) months: \[ \text{Height of Plant B} = 55 + 12t \]

Next, we need to determine when the heights of Plant A and Plant B are equal. To do this, we set the two height expressions equal to each other:

\[ 75 + 10t = 55 + 12t \]

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

1. Subtract \( 10t \) from both sides: \[ 75 = 55 + 2t \]

2. Subtract 55 from both sides: \[ 20 = 2t \]

3. Divide both sides by 2: \[ t = 10 \]

The plants will be the same height after 10 months. Now, we need to find the height at that time. We can substitute \( t = 10 \) into either height expression:

Using Plant A's height: \[ \text{Height of Plant A} = 75 + 10(10) = 75 + 100 = 175 , \text{cm} \]

Using Plant B's height (to verify): \[ \text{Height of Plant B} = 55 + 12(10) = 55 + 120 = 175 , \text{cm} \]

Thus, after 10 months, both plants will be 175 cm tall.