Let’s work through the mathematical problems in the image one by one. I will solve some key examples and give you a detailed explanation. Let me know if you'd like me to address a specific question.

31.

The perimeter of a rectangle is 220 cm. Find the area if its length is 75 cm.

• Formula for perimeter:
  $P = 2(L + B)$
  Substituting $P = 220$ and $L = 75$:
  $220 = 2(75 + B)$
  $110 = 75 + B \implies B = 35 \, \text{cm.}$

• Area of rectangle:
  $\text{Area} = L \times B = 75 \times 35 = 2625 \, \text{cm}^2.$

Answer: (b) 2625 cm².

32.

The breadth of a rectangle is 40 cm. Find the area if its perimeter is 200 cm.

• Formula for perimeter:
  $P = 2(L + B)$
  Substituting $P = 200$ and $B = 40$:
  $200 = 2(L + 40) \implies 100 = L + 40 \implies L = 60 \, \text{cm.}$

• Area:
  $\text{Area} = L \times B = 60 \times 40 = 2400 \, \text{cm}^2.$

Answer: (d) 2400 cm².

33.

The perimeter of a rectangle is 140 cm. If its length is three times its breadth, find the area.

• Let the breadth be $B$. Then length $L = 3B$.
  Perimeter:
  $P = 2(L + B) = 140 \implies 2(3B + B) = 140 \implies 8B = 140 \implies B = 17.5 \, \text{cm.}$
  $L = 3 \times 17.5 = 52.5 \, \text{cm.}$

• Area:
  $\text{Area} = L \times B = 52.5 \times 17.5 = 918.75 \, \text{cm}^2.$

Answer: (a) 918.75 cm².

34.

The length of a rectangle is three more than twice its breadth. If its perimeter is 276 cm, find its length.

• Let the breadth be $B$. Then length $L = 2B + 3$.
  Perimeter:
  $P = 2(L + B) = 276 \implies 2((2B + 3) + B) = 276$
  $2(3B + 3) = 276 \implies 6B + 6 = 276 \implies 6B = 270 \implies B = 45 \, \text{cm.}$
  $L = 2B + 3 = 2(45) + 3 = 93 \, \text{cm.}$

Answer: (d) 93 cm.

35.

The area of a rectangle is 180 cm². If the length of the rectangle is five times its breadth, find the length.

• Let the breadth be $B$. Then length $L = 5B$.
  Area:
  $\text{Area} = L \times B = 180 \implies (5B) \times B = 180 \implies 5B^2 = 180 \implies B^2 = 36 \implies B = 6 \, \text{cm.}$
  $L = 5B = 5 \times 6 = 30 \, \text{cm.}$

Answer: (c) 30 cm.

Would you like me to continue solving the remaining problems? Let me know!

5 Related Questions

1. How is the perimeter formula for a rectangle derived?
2. How do you calculate the area of a rectangle when given the diagonal?
3. Can you explain how length and breadth ratios impact the perimeter?
4. How can you use equations to solve for dimensions of irregular rectangles?
5. What are some real-life applications of rectangle geometry?

Tip:

When solving problems involving rectangles, always start with the perimeter and area formulas. Substitute the known values to form equations for unknown variables.