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Let's break down and solve the case study provided in the image.

### Problem Description:
1. Nikita wants to make a picture frame with rectangular sides whose lengths are \(5x + 3\) and \(3x + 2\). 
2. She wants to put green colored paper inside the frame as shown in the figure.

### Tasks:
a) Find the tape (perimeter) of the border along the sides.
b) Find the area of the green colored paper.
c) Find the cost of making the frame if \(x = 5 \, \text{cm}\) and the cost per centimeter is ₹10.

---

### a) Finding the perimeter:
The perimeter of a rectangle is given by the formula:
\[
P = 2 \times (\text{Length} + \text{Width})
\]
Here, the length is \(5x + 3\) and the width is \(3x + 2\). So, the perimeter is:
\[
P = 2 \times \left( (5x + 3) + (3x + 2) \right)
= 2 \times \left( 5x + 3x + 3 + 2 \right)
= 2 \times (8x + 5)
\]
Thus, the perimeter is:
\[
P = 16x + 10
\]

### b) Finding the area:
The area of the rectangle is given by the formula:
\[
A = \text{Length} \times \text{Width}
\]
Substituting the expressions for length and width:
\[
A = (5x + 3) \times (3x + 2)
\]
Using the distributive property:
\[
A = (5x \times 3x) + (5x \times 2) + (3 \times 3x) + (3 \times 2)
= 15x^2 + 10x + 9x + 6
\]
\[
A = 15x^2 + 19x + 6
\]

### c) Finding the cost of the frame:
Given that \(x = 5 \, \text{cm}\) and the cost per centimeter is ₹10, we can now compute the perimeter and then multiply by the cost.

1. **Perimeter when \(x = 5\):**
\[
P = 16x + 10 = 16(5) + 10 = 80 + 10 = 90 \, \text{cm}
\]

2. **Cost of the frame:**
\[
\text{Cost} = \text{Perimeter} \times \text{Cost per cm}
= 90 \times 10 = ₹900
\]

---

### Final Answers:
a) The perimeter of the frame is \(16x + 10\) cm.  
b) The area of the green colored paper is \(15x^2 + 19x + 6 \, \text{cm}^2\).  
c) The total cost of making the frame is ₹900 when \(x = 5\).

---

Would you like more details or explanations on any step? Here are 5 related questions to expand on this problem:

1. How does changing the value of \(x\) affect the perimeter and area of the frame?
2. What is the physical significance of the perimeter in this case study?
3. How would you calculate the cost if the price per centimeter changed to ₹12?
4. Can you explain how distributive property was used in calculating the area?
5. How would the area formula change if the frame was not rectangular but square?

### Tip:
When solving for expressions like perimeter and area, always simplify the algebraic terms before plugging in the values for \(x\). This makes it easier to calculate later.

Nikita wanted to make a picture frame which is rectangular whose sides are (5x + 3) and (3x + 2). She wants to put green colored paper inside it. a) Find the tape (perimeter) used along the border. b) Find the area of the green colored paper. c) Find the cost of making the frame of picture if x = 5 cm and cost of 1 cm is ₹10.

Learn how to calculate the perimeter, area, and cost of making a rectangular picture frame with sides given by algebraic expressions. Solve for the total cost when x = 5 cm and the cost per centimeter is ₹10. This problem is perfect for Grades 7-9 and covers key geometry and algebra concepts.

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