Chapter 11.9, Problem 27STP

To determine

To calculate the area of the patio.

Answer to Problem 27STP

  $429.7$ square feet.

Explanation of Solution

Given information:

The figure consists of a rectangle and a semicircle. Once the 4 feet wide patio is removed, there would be another set of rectangle and semicircle.

For this sake of calculation, the details of both the sets of rectangle and semicircle is mentioned.

Outer rectangle has length $l_1=30$ feet and breadth $b_1=24$ feet.

Outer semicircle has diameter $d_1=24$ feet. Radius $r_1=\frac{d_1}{2}$

  $=\frac{24}{2}=12$ feet.

Inner rectangle has length $l_2=30-4$

  $=26$ feet and breadth $b_2=24-4-4$

  $=16$ feet.

Inner semicircle has diameter $d_2=24-4-4$

  $=16$ feet.

Formula used: Area of outer inner and rectangle $a_1=l_1\times b_1$ and $a_2=l_2\times b_2$ respectively.

Area of outer and inner semicircle $s_1=\frac{\pi \times r_1{}^2}{2}$ and $s_2=\frac{\pi \times r_2{}^2}{2}$ respectively.

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

Calculation:

To calculate area of the patio, calculate the area of the outer region and subtract the inner region from it. Effectively, 4 feet will be removed from all dimensions while calculating the inner portion.

Both parts have a rectangle and semicircle. This would be found out by adding the areas of the rectangle and semicircle.

Area of outer region.

  $a_1=l_1\times b_1$

Area of the outer rectangle’s formula.

  $=30\times 24$

Applying values in the formula.

  $=720$ square feet.

Area of the outer semi-circle.

  $s_1=\frac{\pi \times r_1{}^2}{2}$

Formula for area of outer semicircle.

  $=\frac{22\times {12}^2}{7\times 2}$

Putting the values in the formula.

  $=\frac{11\times 144}{7}$

Simplifying the same.

  $=226.29$ square feet.

Now, area of the outer region

  $A_1=a_1+s_1$

Add the areas of the outer portion

  $=720+226.29$

Numeric values of the respective shapes.

  $=946.29$ square feet.

Area of inner region.

  $a_2=l_2\times b_2$

Area of the inner rectangle’s formula.

  $=26\times 16$

Applying values in the formula.

  $=416$ square feet.

Area of the inner semi-circle.

  $s_2=\frac{\pi \times r_2{}^2}{2}$

Formula for area of inner semicircle.

  $=\frac{22\times 8^2}{7\times 2}$

Putting the values in the formula.

  $=\frac{11\times 64}{7}$

Simplifying the same.

  $=100.57$ square feet.

Now, area of the inner region

  $A_2=a_2+s_2$

Add the areas of the outer portion

  $=416+100.57$

Numeric values of the respective shapes.

  $=516.57$ square feet.

Area of the patio would be the difference between the areas of the outer and inner regions.

This is given by.

  $=A_1-A_2$

Areas of the two regions.

  $=946.29-516.57$

Subtracting the same.

  $=429.72$

Rounding off to the nearest tenth.

  $=429.7$ square feet.