Q14:

A circular flower garden has an area of $314$ m$^2$. A sprinkler at the centre of the garden can cover an area that has a radius of $12$ m. Will the sprinkler water the entire garden? (Take $\pi = 3.14$)

Given Variables & Initial Setup

To determine whether the sprinkler can cover the entire garden, we must analyze the spatial properties of both the garden and the sprinkler's reach. We are provided with the following parameters:

• Area of the circular flower garden, $A_g = 314 \text{ m}^2$
• Radius of the sprinkler's coverage, $r_s = 12 \text{ m}$
• Approximation for Pi, $\pi = 3.14$

Step 1: Determining the Radius of the Garden

The garden is circular. We can find its radius ($r_g$) by utilizing the standard formula for the area of a circle [Per the geometric theorem for circular area: $A = \pi r^2$].

Setting up the equation with the given area:

$A_g = \pi r_g^2$

Substituting the known values into the equation:

$314 = 3.14 \times r_g^2$

Isolating $r_g^2$ by dividing both sides by $3.14$ [By the Division Property of Equality]:

$r_g^2 = \frac{314}{3.14}$

$r_g^2 = 100$

Taking the principal square root of both sides to find the radius [Since distance must be a non-negative real number]:

$r_g = \sqrt{100} = 10 \text{ m}$

Step 2: Comparing the Radii

For a sprinkler located at the exact center of a circular garden to water the entire area, the radius of the sprinkler's coverage ($r_s$) must be greater than or equal to the radius of the garden ($r_g$).

$r_s = 12 \text{ m}$

$r_g = 10 \text{ m}$

Comparing the two values:

$12 \text{ m} > 10 \text{ m} \implies r_s > r_g$

Because the sprinkler throws water $2 \text{ m}$ beyond the boundary of the garden, it will successfully cover the entire garden.

Step 3: Verification via Area Comparison (Rigorous Proof)

Alternatively, we can verify this by calculating the total area the sprinkler can cover ($A_s$) and comparing it to the garden's area ($A_g$).

$A_s = \pi r_s^2$

$A_s = 3.14 \times (12)^2$

$A_s = 3.14 \times 144$

$A_s = 452.16 \text{ m}^2$

Since $452.16 \text{ m}^2 > 314 \text{ m}^2$, the area covered by the sprinkler strictly encompasses the area of the garden.

Visual Representation of Coverage

The following scale diagram illustrates the garden (green) and the sprinkler's coverage area (blue dashed line). The scale is $1 \text{ m} = 10 \text{ units}$.

Final Solution: Yes, the sprinkler will water the entire garden. The radius of the garden is $10 \text{ m}$, which is strictly less than the $12 \text{ m}$ radius covered by the sprinkler.