Question
A tree in Elsa's backyard is 46 centimeters high. The tree grows 6 centimeters each month. Within the first month of planting it, Elsa trimmed 2 centimeters off its height. She has not trimmed the tree since then. How long has the tree been growing?
(a) Write an equation to help answer the question above.
Choose the correct form below and fill in the blanks using only , , and .
(The variable is used to represent the number of months.)
(b) Write an expression to help answer the question above.
Choose the correct form below and fill in the blanks using only , , and .
(c) How long has Elsa's tree been growing?
(a) Write an equation to help answer the question above.
Choose the correct form below and fill in the blanks using only , , and .
(The variable is used to represent the number of months.)
(b) Write an expression to help answer the question above.
Choose the correct form below and fill in the blanks using only , , and .
(c) How long has Elsa's tree been growing?
Answers
GPT-4o mini
To determine how long the tree has been growing, let's define the variable and set up our equations based on the problem statement.
### (a) Write an equation to help answer the question above.
Let's use \( t \) to represent the number of months the tree has been growing since it was planted. At the time of planting, the tree was 46 cm high. After one month, it was 46 cm - 2 cm = 44 cm high. The tree then grows 6 cm for each subsequent month. Therefore, the equation for the height of the tree after \( t \) months can be set up as follows:
\[ \text{Height} = 44 + 6(t - 1) \]
Where \( t - 1 \) represents the months it has grown after the first month when it was trimmed. Then, we set this equal to the overall height of the tree:
\[ 44 + 6(t - 1) = 46 \]
So the completed form for (a) is:
\[ 44 + 6(t - 1) = 46 \]
### (b) Write an expression to help answer the question above.
The expression for the height of the tree after \( t \) months is:
\[ 44 + 6(t - 1) \]
This represents the height of the tree at any month \( t \).
So the completed form for (b) is:
\[ 44 + 6(t - 1) \]
### (c) How long has Elsa's tree been growing?
Now, we will solve the equation from part (a):
Starting with:
\[ 44 + 6(t - 1) = 46 \]
First, subtract 44 from both sides:
\[ 6(t - 1) = 2 \]
Next, divide both sides by 6:
\[ t - 1 = \frac{2}{6} \]
\[ t - 1 = \frac{1}{3} \]
Now add 1 to both sides:
\[ t = 1 + \frac{1}{3} \]
\[ t = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \]
This means the tree has been growing for \( \frac{4}{3} \) months, which is equivalent to 1 month and 10 days.
Thus, in conclusion:
### (c) The tree has been growing for approximately 1 month and 10 days.
### (a) Write an equation to help answer the question above.
Let's use \( t \) to represent the number of months the tree has been growing since it was planted. At the time of planting, the tree was 46 cm high. After one month, it was 46 cm - 2 cm = 44 cm high. The tree then grows 6 cm for each subsequent month. Therefore, the equation for the height of the tree after \( t \) months can be set up as follows:
\[ \text{Height} = 44 + 6(t - 1) \]
Where \( t - 1 \) represents the months it has grown after the first month when it was trimmed. Then, we set this equal to the overall height of the tree:
\[ 44 + 6(t - 1) = 46 \]
So the completed form for (a) is:
\[ 44 + 6(t - 1) = 46 \]
### (b) Write an expression to help answer the question above.
The expression for the height of the tree after \( t \) months is:
\[ 44 + 6(t - 1) \]
This represents the height of the tree at any month \( t \).
So the completed form for (b) is:
\[ 44 + 6(t - 1) \]
### (c) How long has Elsa's tree been growing?
Now, we will solve the equation from part (a):
Starting with:
\[ 44 + 6(t - 1) = 46 \]
First, subtract 44 from both sides:
\[ 6(t - 1) = 2 \]
Next, divide both sides by 6:
\[ t - 1 = \frac{2}{6} \]
\[ t - 1 = \frac{1}{3} \]
Now add 1 to both sides:
\[ t = 1 + \frac{1}{3} \]
\[ t = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \]
This means the tree has been growing for \( \frac{4}{3} \) months, which is equivalent to 1 month and 10 days.
Thus, in conclusion:
### (c) The tree has been growing for approximately 1 month and 10 days.