https://file.aiquickdraw.com/mathbot/file/1726155660141y1qwg90v.jpg

From the image, the question states:

- The area of the vegetable garden is 80 m².
- A fence with a 1 m wide iron net was built around the green land.
- The shape in the diagram suggests a rectangular garden.
- The dimensions of the garden are marked as \( x + 2 \) and \( x \) meters.

The goal is to find how many meters of wire (presumably the perimeter) are needed for the fence.

### Step-by-step solution:

1. **Area of the garden**: 
   The area of the rectangle is given by the formula for the area of a rectangle: 
   \[
   \text{Area} = \text{length} \times \text{width}
   \]
   Here, the length is \( x + 2 \) and the width is \( x \). Therefore, we have:
   \[
   (x + 2) \times x = 80
   \]
   Expanding this equation:
   \[
   x^2 + 2x = 80
   \]
   Rearranging:
   \[
   x^2 + 2x - 80 = 0
   \]
   Now we solve this quadratic equation using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   where \( a = 1 \), \( b = 2 \), and \( c = -80 \).

   Plugging in the values:
   \[
   x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-80)}}{2(1)}
   \]
   \[
   x = \frac{-2 \pm \sqrt{4 + 320}}{2}
   \]
   \[
   x = \frac{-2 \pm \sqrt{324}}{2}
   \]
   \[
   x = \frac{-2 \pm 18}{2}
   \]
   So, we have two possible solutions:
   \[
   x = \frac{-2 + 18}{2} = 8 \quad \text{or} \quad x = \frac{-2 - 18}{2} = -10
   \]

   Since the width cannot be negative, we take \( x = 8 \).

2. **Dimensions of the garden**:
   - Width: \( x = 8 \) meters
   - Length: \( x + 2 = 10 \) meters

3. **Perimeter**:
   The perimeter of a rectangle is given by:
   \[
   \text{Perimeter} = 2(\text{length} + \text{width}) = 2(10 + 8) = 2(18) = 36 \, \text{meters}
   \]

Thus, **36 meters of wire** are needed to fence the garden.

---

Do you have any questions or would you like more details?

### Related questions:
1. How do you solve more complex quadratic equations?
2. What are some other methods to calculate the perimeter of irregular shapes?
3. How does the quadratic formula work for non-integer solutions?
4. Could this problem be solved using a different approach?
5. What would happen if the garden had a different shape, like a circle or triangle?

### Tip: 
Always ensure that the dimensions of any physical object make sense within the context of the problem. Negative values often indicate extraneous solutions.

The area of the vegetable garden in the form shown in the figure was 80 m². A fence with a 1 m wide iron net was built around the green land. How many meters of wire are needed?

Learn how to calculate the perimeter of a vegetable garden using a quadratic equation. This problem involves solving for the dimensions of a rectangular garden with an area of 80 m² and determining the length of wire needed for fencing.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation