5. An inground rectangular pool has a concrete pathway surrounding the pool. If the pool is 16 feet by 32 feet and the entire area of the pool including the wallkway is $$ 924 \mathrm{ft}^{2} $$, find the width of the walkway. $$ \begin{array}{l} ext { 6. Given th } \ ext { shaded } \ ext { of the in }\end{array} $$

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5. An inground rectangular pool has a concrete pathway surrounding the pool. If the pool is 16 feet by 32 feet and the entire area of the pool including the wallkway is $$ 924 \mathrm{ft}^{2} $$, find the width of the walkway. $$ \begin{array}{l}	ext { 6. Given th } \ 	ext { shaded } \ 	ext { of the in }\end{array} $$

UpStudy AI · Accepted Answer

Let $$ x $$ be the width of the walkway in feet.

1. The pool has dimensions $$ 16 $$ ft by $$ 32 $$ ft, so the area of the pool is
   $$
   16 \times 32 = 512 \text{ ft}^2.
   $$

2. With the walkway, the overall dimensions become 
   $$
   (16+2x) \text{ ft} \quad \text{and} \quad (32+2x) \text{ ft}.
   $$
   Therefore, the total area including the walkway is
   $$
   (16+2x)(32+2x) = 924 \text{ ft}^2.
   $$

3. Expanding the product:
   $$
   (16+2x)(32+2x) = 16 \times 32 + 16(2x) + 32(2x) + (2x)^2 = 512 + 32x + 64x + 4x^2.
   $$
   Combining like terms gives:
   $$
   4x^2 + 96x + 512 = 924.
   $$

4. Subtract $$ 924 $$ from both sides to set the equation equal to zero:
   $$
   4x^2 + 96x + 512 - 924 = 0 \quad \Longrightarrow \quad 4x^2 + 96x - 412 = 0.
   $$

5. Divide the entire equation by 2 to simplify:
   $$
   2x^2 + 48x - 206 = 0.
   $$

6. Use the quadratic formula:
   $$
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
   $$
   where $$ a = 2 $$, $$ b = 48 $$, and $$ c = -206 $$.

7. Calculate the discriminant:
   $$
   b^2 - 4ac = 48^2 - 4(2)(-206) = 2304 + 1648 = 3952.
   $$

8. Write the solution:
   $$
   x = \frac{-48 \pm \sqrt{3952}}{4}.
   $$

9. Simplify $$\sqrt{3952}$$:
   $$
   \sqrt{3952} = \sqrt{16 \times 247} = 4\sqrt{247}.
   $$
   So,
   $$
   x = \frac{-48 \pm 4\sqrt{247}}{4} = -12 \pm \sqrt{247}.
   $$

10. Since $$ x $$ must be positive, we choose:
    $$
    x = -12 + \sqrt{247}.
    $$

Thus, the width of the walkway is
$$
\boxed{x = \sqrt{247} - 12 \text{ ft}},
$$
which is approximately
$$
\sqrt{247} \approx 15.72 \quad \Longrightarrow \quad x \approx 15.72 - 12 = 3.72 \text{ ft}.
$$
The width of the walkway is approximately 3.72 feet.

An inground rectangular pool has a concrete
pathway surrounding the pool. If the pool is 16
feet by 32 feet and the entire area of the pool
including the wallkway is , find the width
of the walkway.

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An inground rectangular pool has a concrete pathway surrounding the pool. If the pool is 16 feet by 32 feet and the entire area of the pool including the wallkway is , find the width of the walkway.