https://file.aiquickdraw.com/mathbot/file/17291066194141moe83f6.jpg

We are given the problem where the total shaded area is \( 75 \, \text{m}^2 \), and we need to find the area of the smaller rectangle. Here is the approach:

### Step-by-step solution:
1. **Outer rectangle dimensions**:  
   The dimensions of the outer green rectangle are:
   - Length = \( x + 4 \, \text{m} \)
   - Width = \( 2x + 3 \, \text{m} \)

   So, the area of the outer rectangle is:
   \[
   A_{\text{outer}} = (x + 4)(2x + 3)
   \]

2. **Inner (smaller) rectangle dimensions**:  
   The dimensions of the smaller rectangle (which is white in the image) are:
   - Length = \( x + 2 \, \text{m} \)
   - Width = \( x + 1 \, \text{m} \)

   So, the area of the inner rectangle is:
   \[
   A_{\text{inner}} = (x + 2)(x + 1)
   \]

3. **Shaded area**:  
   The shaded area is the difference between the outer rectangle and the inner rectangle, and we know it equals \( 75 \, \text{m}^2 \):
   \[
   A_{\text{shaded}} = A_{\text{outer}} - A_{\text{inner}} = 75
   \]

4. **Set up the equation**:  
   Expanding both expressions:
   - \( A_{\text{outer}} = (x + 4)(2x + 3) = 2x^2 + 3x + 8x + 12 = 2x^2 + 11x + 12 \)
   - \( A_{\text{inner}} = (x + 2)(x + 1) = x^2 + x + 2x + 2 = x^2 + 3x + 2 \)

   Subtracting these:
   \[
   A_{\text{shaded}} = (2x^2 + 11x + 12) - (x^2 + 3x + 2) = x^2 + 8x + 10
   \]

   Setting it equal to 75:
   \[
   x^2 + 8x + 10 = 75
   \]

5. **Solve the quadratic equation**:  
   Rearrange the equation:
   \[
   x^2 + 8x - 65 = 0
   \]

   Solve this quadratic equation using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   where \( a = 1 \), \( b = 8 \), and \( c = -65 \).

   Substituting the values:
   \[
   x = \frac{-8 \pm \sqrt{8^2 - 4(1)(-65)}}{2(1)} = \frac{-8 \pm \sqrt{64 + 260}}{2} = \frac{-8 \pm \sqrt{324}}{2} = \frac{-8 \pm 18}{2}
   \]

   So, \( x = \frac{-8 + 18}{2} = \frac{10}{2} = 5 \).

6. **Area of the smaller rectangle**:  
   Now that \( x = 5 \), we substitute this into the expression for the area of the smaller rectangle:
   \[
   A_{\text{inner}} = (x + 2)(x + 1) = (5 + 2)(5 + 1) = 7 \times 6 = 42 \, \text{m}^2
   \]

### Final Answer:
The area of the smaller rectangle is \( 42 \, \text{m}^2 \).

---

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

Here are some related questions you might consider:
1. How would the problem change if the shaded area was different?
2. What happens if we change the dimensions of the inner rectangle?
3. Can we solve for \( x \) if the outer rectangle has different dimensions?
4. How does the quadratic formula handle cases where there are no real roots?
5. Can the shaded area ever be negative in a similar setup?

**Tip**: Always check that your calculated areas make sense in the context of the problem to avoid logical errors!

The shaded area is 75 m². Work out the area of the smaller rectangle.

Learn how to calculate the area of a smaller rectangle within a larger figure by solving a quadratic equation. This problem involves finding dimensions using algebra and geometry concepts, making it suitable for Grades 8-10 students. Includes step-by-step solutions with a quadratic equation application.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation