A swimming pool is in the form of a semicircular-sided figure. Its width is 16 ft and the parallel portions of the sides are each 16
f… What is the area of a 3 -fi-wide walk surrounding the pool?

Question

A swimming pool is in the form of a semicircular-sided figure. Its width is 16 ft and the parallel portions of the sides are each 16 f.. What is the area of a 3 -fi-wide walk surrounding the pool?

UpStudy AI · Accepted Answer

To find the area of the 3-ft-wide walk surrounding the swimming pool, we first need to determine the dimensions of the pool itself.

The pool is described as having a semicircular side and two parallel sides, each 16 ft long. This means the width of the pool is 16 ft, and the semicircular part has a diameter of 16 ft. Therefore, the radius $$ r $$ of the semicircle is:

$$
r = \frac{16}{2} = 8 \text{ ft}
$$

Next, we can calculate the area of the pool. The area of the semicircular part of the pool is given by the formula for the area of a circle, divided by 2:

$$
\text{Area of semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (8^2) = \frac{1}{2} \pi (64) = 32\pi \text{ ft}^2
$$

The area of the rectangular part of the pool (the two parallel sides) can be calculated as follows:

$$
\text{Area of rectangle} = \text{width} \times \text{height} = 16 \text{ ft} \times 16 \text{ ft} = 256 \text{ ft}^2
$$

Thus, the total area of the pool is:

$$
\text{Total area of pool} = \text{Area of semicircle} + \text{Area of rectangle} = 32\pi + 256 \text{ ft}^2
$$

Now, we need to calculate the area of the walk surrounding the pool. The walk is 3 ft wide, which means we need to add 3 ft to each side of the pool.

The new dimensions of the pool including the walk will be:

- The width of the pool plus the walk on both sides: $$ 16 + 2 \times 3 = 22 \text{ ft} $$
- The height of the pool (the length of the parallel sides) plus the walk on both sides: $$ 16 + 2 \times 3 = 22 \text{ ft} $$

The radius of the semicircle including the walk will be:

$$
r + 3 = 8 + 3 = 11 \text{ ft}
$$

Now we can calculate the area of the larger semicircle (including the walk):

$$
\text{Area of larger semicircle} = \frac{1}{2} \pi (11^2) = \frac{1}{2} \pi (121) = 60.5\pi \text{ ft}^2
$$

And the area of the larger rectangle (including the walk) is:

$$
\text{Area of larger rectangle} = 22 \text{ ft} \times 22 \text{ ft} = 484 \text{ ft}^2
$$

Thus, the total area of the larger pool (including the walk) is:

$$
\text{Total area of larger pool} = \text{Area of larger semicircle} + \text{Area of larger rectangle} = 60.5\pi + 484 \text{ ft}^2
$$

Finally, the area of the walk itself is the area of the larger pool minus the area of the original pool:

$$
\text{Area of walk} = \text{Total area of larger pool} - \text{Total area of pool}
$$

Calculating this gives:

$$
\text{Area of walk} = (60.5\pi + 484) - (32\pi + 256)
$$

Simplifying this:

$$
\text{Area of walk} = (60.5\pi - 32\pi) + (484 - 256) = 28.5\pi + 228 \text{ ft}^2
$$

Thus, the area of the 3-ft-wide walk surrounding the pool is:

$$
\text{Area of walk} \approx 28.5 \times 3.14 + 228 \approx 89.49 + 228 \approx 317.49 \text{ ft}^2
$$

So, the area of the walk surrounding the pool is approximately $$ 317.49 \text{ ft}^2 $$.
The area of the 3-ft-wide walk surrounding the pool is approximately $$ 317.49 \text{ ft}^2 $$.

A swimming pool is in the form of a semicircular-sided figure. Its width is 16 ft and the parallel portions of the sides are each 16
f… What is the area of a 3 -fi-wide walk surrounding the pool?

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A swimming pool is in the form of a semicircular-sided figure. Its width is 16 ft and the parallel portions of the sides are each 16 f… What is the area of a 3 -fi-wide walk surrounding the pool?

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A swimming pool is in the form of a semicircular-sided figure. Its width is 16 ft and the parallel portions of the sides are each 16
f… What is the area of a 3 -fi-wide walk surrounding the pool?