Question

Around a square pool there is a footpath of width 2m. If the area of the footpath $\frac{5}{4}$ times that of the pool. Find the area of the pool.

Answer 1

Let x m be the side of the square pool. Then, its area is x² m².
Now, the width of the footpath around the square pool is 2 m. Thus, the side of the square pool, including the footpath is (x + 2 + 2) = (x + 4) m and its area is (x + 4)² m².
Now, we draw a figure of the square pool with its footpath as shown below.

From the figure, we observe that area of the footpath is
ar(PQRS) – ar(ABCD) = (x + 4)² – x² = x² + 8x + 16 – x² = 8x + 16
Now, by the given condition, we get:
8x + 16 = $\frac{5}{4}{x}^2$
On multiplying both sides by 4, we get:
32x + 64 = 5x²
$\Rightarrow$ 5x² – 32x – 64 = 0
On splitting the middle term –32x as 8x – 40x, we get:
5x² + 8x – 40x – 64 = 0
$\Rightarrow$ x(5x + 8) – 8(5x + 8) = 0
$\Rightarrow$ (5x + 8)(x – 8) = 0
$\Rightarrow$ 5x + 8 = 0 or x – 8 = 0
$\Rightarrow$ x = –$\frac{8}{5}$ or x = 8
Since x is the side of the square pool, which cannot be negative, we get x = 8
Thus, the area of the square pool is (8)² = 64 sq. m.