Question

Governing council of a local public development authority of Dehradun decided to build an adventurous playground on the top of a hill, which will have adequate space for parking. After survey, it was decided to build a rectangular playground, with a semicircular area allotted for parking at one end of the playground. The length and breadth of the rectangular playground are 14 units and 7 units, respectively. There are two quadrants of radius 2 units on one side for special seats.

Based on the given information, answer the following questions.

1. What is the total perimeter of the parking area?
2. What is the total area of the parking and the two quadrants?
3. What is the ratio of the area of the playground to the area of the parking area?
4. Find the cost of fencing the playground and parking area at the rate of ₹ 2 per unit.

Answer 1

i. Perimeter of the parking area (semicircle)

Breadth of the rectangle = 7 units, so radius of the semicircle \(r = \dfrac{7}{2} = 3.5\) units.

\[\text{Perimeter} = \pi r + 2r\]

\[= \frac{22}{7} \times \frac{7}{2} + 2 \times \frac{7}{2}\]

\[= 11 + 7\]

\[= 18\ \text{units}\]

ii. Total area of the parking and the two quadrants

Area of parking (semicircle):

\[\frac{1}{2}\pi r^2 = \frac{1}{2} \times \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2}\]

\[= 19.25\ \text{sq units}\]

Area of two quadrants of radius 2 units:

\[2 \times \frac{1}{4}\pi r^2 = 2 \times \frac{1}{4} \times \frac{22}{7} \times 2 \times 2\]

\[= \frac{44}{7} \approx 6.29\ \text{sq units} \] 

\[\text{Total area} = 19.25 + 6.29\]

\[= 25.54\ \text{sq units}\]

iii. Ratio of the area of the playground to the area of the parking area

Area of rectangular playground \(= 14 \times 7 = 98\) sq units.

\[\text{Ratio} = \frac{98}{19.25}\]

\[= \frac{98}{\frac{77}{4}}\]

\[= \frac{98 \times 4}{77}\]

\[= \frac{56}{11}\]

\[= 56 : 11\]

iv. Cost of fencing the playground and parking area

Total perimeter to be fenced:

\[2(14 + 7) + \frac{22}{7} \times \frac{7}{2} = 42 + 11\]

\[= 53\ \text{units} \] 

\[\text{Cost} = 53 \times 2\]

\[= ₹,106 \]