The problem describes a flower bed consisting of a semicircle of radius 14m adjoining a rectangular part ABCD. Pebbles have been scattered in the two shaded rectangular parts outside the flower bed. We are given $\pi = \frac{22}{7}$ and asked to find: (i) The length BC of the rectangular part of the flower bed. (ii) The area of the semicircular part of the flower bed. (iii) The length AB of the rectangular part, given that the area of the semicircular part is equal to the sum of the areas of the two parts in which pebbles have been scattered. (iv) The perimeter of the whole flower bed and then find the length of a rectangle that has the same perimeter as the flower bed and breadth equal to the diameter of the semicircle.

1. Problem Description

The problem describes a flower bed consisting of a semicircle of radius 14m adjoining a rectangular part ABCD. Pebbles have been scattered in the two shaded rectangular parts outside the flower bed. We are given $\pi = \frac{22}{7}$ and asked to find:

(i) The length BC of the rectangular part of the flower bed.

(ii) The area of the semicircular part of the flower bed.

(iii) The length AB of the rectangular part, given that the area of the semicircular part is equal to the sum of the areas of the two parts in which pebbles have been scattered.

(iv) The perimeter of the whole flower bed and then find the length of a rectangle that has the same perimeter as the flower bed and breadth equal to the diameter of the semicircle.

2. Solution Steps

(i) Finding the length BC:

From the diagram, the radius of the semicircle is 14m. This radius is also equal to AD. Since ABCD is a rectangle, BC = AD. Therefore, BC = 14m.

(ii) Finding the area of the semicircular part:

The radius of the semicircle is given as 14m. The area of a full circle is given by the formula $A = \pi r^2$, where r is the radius. Since we have a semicircle, we need to divide the area of the full circle by

2.

$A_{semicircle} = \frac{1}{2} \pi r^2$

$A_{semicircle} = \frac{1}{2} \times \frac{22}{7} \times (14)^2$

$A_{semicircle} = \frac{1}{2} \times \frac{22}{7} \times 196$

$A_{semicircle} = \frac{1}{2} \times 22 \times 28$

$A_{semicircle} = 11 \times 28$

$A_{semicircle} = 308 \, m^2$

(iii) Finding the length AB:

The area of the semicircle is equal to the sum of the areas of the two shaded rectangular parts.

Area of each shaded rectangle = 7 * AB

Area of two shaded rectangles = 2 * 7 * AB = 14 * AB

We are given that the area of the semicircle is equal to the area of the two shaded parts.

$308 = 14 \times AB$

$AB = \frac{308}{14} = 22 \, m$

(iv) Finding the perimeter of the whole flower bed:

The perimeter of the flower bed is the sum of the lengths AB, BC, CD, and the arc length of the semicircle.

AB = 22m (calculated in (iii))

BC = 14m (calculated in (i))

CD = 22m (same as AB)

Arc length of semicircle = $\frac{1}{2} \times 2 \pi r = \pi r = \frac{22}{7} \times 14 = 22 \times 2 = 44 \, m$

Perimeter of flower bed = AB + BC + CD + Arc length

Perimeter = $22 + 14 + 22 + 44 = 102 \, m$

Now, we need to find the length of a rectangle with the same perimeter as the flower bed (102m) and breadth equal to the diameter of the semicircle.

Breadth = Diameter = 2 * radius = 2 * 14 = 28m

Perimeter of rectangle = 2(length + breadth)

$102 = 2(length + 28)$

$51 = length + 28$

$length = 51 - 28 = 23 \, m$

3. Final Answer

(i) BC = 14m

(ii) Area of semicircle = 308 $m^2$

(iii) AB = 22m

(iv) Perimeter of flower bed = 102m, Length of rectangle = 23m