https://file.aiquickdraw.com/mathbot/file/1726279007396g73mso55.jpg

Let's break down the problem and find a solution step by step:

### Problem Statement:
- Simon has two trees: a **maple tree** and an **oak tree**.
- **Maple tree**:
  - Initial height: 5 cm
  - Growth rate: 3 cm per month
- **Oak tree**:
  - Initial height: 10 cm
  - Growth rate: 2 cm per month
- The question asks: After how many months will the two trees have the same height?

### Step 1: Write Equations
Let \( t \) be the number of months.

1. **Height of the Maple Tree after \( t \) months**:
   - The maple tree starts at 5 cm and grows by 3 cm per month.
   - Equation: \( H_{\text{maple}} = 5 + 3t \)

2. **Height of the Oak Tree after \( t \) months**:
   - The oak tree starts at 10 cm and grows by 2 cm per month.
   - Equation: \( H_{\text{oak}} = 10 + 2t \)

### Step 2: Set Up the System of Equations
We want to find the time \( t \) when the two trees have the same height. That means:
\[
5 + 3t = 10 + 2t
\]

### Step 3: Solve for \( t \)
Let's solve the equation for \( t \).

\[
5 + 3t = 10 + 2t
\]
Subtract \( 2t \) from both sides:
\[
5 + t = 10
\]
Subtract 5 from both sides:
\[
t = 5
\]

### Step 4: Solution Interpretation
The two trees will be the same height after **5 months**.

### Step 5: Graphing
For graphing purposes:
- The **maple tree**'s height equation is \( H_{\text{maple}} = 5 + 3t \), which is a straight line starting at 5 and with a slope of 3.
- The **oak tree**'s height equation is \( H_{\text{oak}} = 10 + 2t \), which is a straight line starting at 10 and with a slope of 2.

Both lines will intersect at \( t = 5 \), and the height at that point will be:
\[
H_{\text{maple}} = H_{\text{oak}} = 5 + 3(5) = 20 \, \text{cm}
\]

The two trees will be 20 cm tall after 5 months.

---

Would you like to see a detailed graph or have any additional questions?

Here are 5 related questions:
1. How would the solution change if the maple tree grew faster than 3 cm per month?
2. What if both trees had the same initial height but different growth rates?
3. How would the graph look if one of the trees had no growth over time?
4. Can we model the tree's height using a quadratic equation if growth is not linear?
5. What happens to the system if one of the trees decreases in height over time?

**Tip:** When solving linear systems, finding where two lines intersect gives you the solution to the problem.

Simon has been planting young trees in his garden. The maple tree that is 5 centimeters tall is growing 3 centimeters per month, whereas the oak tree that is 10 centimeters tall is growing 2 centimeters per month. In a few months, the two trees will be the same height. How long will that take? Write a system of equations, graph them, and type the solution.

Simon planted a maple and an oak tree, and we need to determine when their heights will be equal. Using systems of linear equations, this problem provides a step-by-step guide to finding the solution, graphing the equations, and interpreting the results. Suitable for Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation