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Let's go through the questions from the image step by step:

### Given:
- The original patio is a rectangle with:
  - Length = 5 meters
  - Width = 4 meters

The length and width will be extended by **x** meters to create more space.

---

### Part a)
**Task**: Draw a diagram to represent the new patio.

- The new dimensions of the patio will be:
  - Length: $ 5 + x $ meters
  - Width: $ 4 + x $ meters
  
To represent the new patio, simply sketch a rectangle labeled with the original dimensions (5m by 4m) and an extension labeled by $ x $ meters on both the length and width.

---

### Part b)
**Task**: Write a simplified expression for the area of the new patio, giving your answer in expanded form.

The area of a rectangle is given by:
\[
\text{Area} = \text{Length} \times \text{Width}
\]

For the new patio:
\[
\text{Area} = (5 + x)(4 + x)
\]

Expanding this expression:
\[
(5 + x)(4 + x) = 5 \times 4 + 5 \times x + x \times 4 + x \times x
\]
\[
= 20 + 5x + 4x + x^2
\]
\[
= x^2 + 9x + 20
\]

So, the simplified expression for the area is:
\[
x^2 + 9x + 20 \, \text{square meters}.
\]

---

### Part c)
#### i) 
**Task**: The owners decide to extend both the length and width by 3 meters. Use the expression from part (b) to calculate the area of the new patio.

Substitute $ x = 3 $ into the expression $ x^2 + 9x + 20 $:

\[
\text{Area} = 3^2 + 9 \times 3 + 20
\]
\[
= 9 + 27 + 20
\]
\[
= 56 \, \text{square meters}.
\]

So, the area of the new patio is 56 square meters.

---

#### ii) 
**Task**: If materials cost $50 per square meter, calculate the total cost of the project.

The cost of materials is given by:
\[
\text{Total cost} = \text{Area} \times \text{Cost per square meter}
\]

Using the area of 56 square meters and cost of $50 per square meter:
\[
\text{Total cost} = 56 \times 50 = 2800 \, \text{dollars}.
\]

---

Would you like further details on any of the steps?

### Related Questions:
1. How would the area change if the owners extended by 4 meters instead of 3?
2. What if the material cost was $60 per square meter?
3. How does the quadratic expression in part (b) help in predicting the area for various values of $ x $?
4. What would be the perimeter of the new patio when $ x = 3 $?
5. Could the patio's shape be optimized to reduce material costs?

### Tip:
When expanding products like $ (a + b)(c + d) $, always apply the distributive property carefully to avoid missing terms.

The back patio of a house is rectangular in shape, with a length of 5 meters and a width of 4 meters. The patio is to be knocked down and rebuilt completely, and both the length and the width are to be extended to create more space. The owners are unsure by how much they should extend the dimensions or how much the new project will cost. They decide to extend both the length and width by 'x' meters to explore their options.

Learn how to calculate the area of a patio when its length and width are extended by a variable amount 'x'. This problem involves algebraic concepts, including expanding quadratic expressions and using the formula for the area of a rectangle. Suitable for students in Grades 7-9.

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