Chapter 11.9, Problem 11PPS

To determine

To calculate the area of the figure.

Answer to Problem 11PPS

  $257.1$ square feet.

Explanation of Solution

Given information:

The figure is can be divided into a square and four equal semicircles.

The length of the side of the square $=a$ and twice the radius $=r$ of the semicircle adds up to 20 feet.

That is, $a+2r=20$ .

Formula used: Area of a square $s=a\times a$ where ‘a’ is the side of the square.

Area of semicircle $c=\frac{\pi \times r^2}{2}$ where ‘r’ is the radius.

For calculations, $\pi =\frac{22}{7}$ has been used.

Calculation:

On noticing the placement of the semicircles and the square, it is observed that the side of the square is twice the radius of the semicircle.

That is,

  $a=2\times r$ where ‘a’ is the side of the square and ‘r’ is the radius of the semicircle.

Thus, it can be said that

  $a+2r=20$

Is the same as

  $2r+2r=20$

Replacing the value of ‘a’ by twice ‘r’.

  $=>4r=20$

On further simplification.

  $=>r=\frac{20}{4}$

On dividing the right-hand side numbers.

  $=>r=5$ feet.

And, ‘a’ would be equal to.

  $a=2\times 5$

Replacing the value of ‘r’.

  $a=10$ feet.

Now, find the area of the shapes that constitute the composite figure mentioned in the problem.

Area of the square.

  $s=a\times a$

Putting the value of ‘a’ in this formula.

  $=10\times 10$

On solving this.

  $=100$ square feet.

Area of semicircle

  $c=\frac{\pi \times r^2}{2}$

Formula for area of semicircle.

Now, since all the 4 semicircles have the same radius, multiple the area of one semicircle with 4. This will get the value of all the 4 semicircles.

  $=4\times \frac{\pi \times r^2}{2}$

Putting the values in the formula

  $=4\times \frac{22}{7}\times \frac{5^2}{2}$

Simplify the expression

  $=4\times \frac{22}{7}\times \frac{25}{2}$

On further calculation,

  $=\frac{2200}{14}$

Dividing the numerator by the denominator.

  $=157.14$ square feet.

Now, add the two areas to get the area of the given figure.

  $=s+c$

Putting values of the two areas.

  $=100+157.14$

It becomes

  $=257.14$

After rounding this to the nearest tenth

  $=257.1$ square feet.