Chapter 11.9, Problem 20PPS

To determine

To calculate the area of the shaded region.

Answer to Problem 20PPS

  $135$ square feet.

Explanation of Solution

Given information:

The figure consists of a trapezoid along with a non-shaded triangle.

Trapezoid has sides $a=18$ feet, $b=8$ and the height would be calculated using Pythagoras theorem.

Triangle of $b=8$ feet and height would be calculated using Pythagoras theorem.

Formula used: Area of trapezoid $t=\frac{a+b}{2}\times p$ , where ‘a’ and ‘b’ are parallel sides while ‘p’ is the perpendicular height.

Area of triangle $s=\frac{1}{2}\times b\times p$ . Where ‘b’ is the base and ‘p’is the perpendicular height.

Pythagoras theorem will be used for finding the perpendicular height using $p^2+b^2=h^2$ .

Here ‘p’ is the perpendicular height, ‘b’ is the base and ‘h’ is the hypotenuse of the right-angled triangle.

Calculation:

Calculate the height of the given figure first. This will enable to determine the overall area as well as the area of the non-shaded right-angle triangle present within the figure.

  $p^2+b^2=h^2$

Using Pythagoras theorem.

  $p^2=h^2-b^2$

Since the value of the perpendicular height is needed, arrange the equation with keeping ‘p’ in the left-hand side.

  $p=\sqrt{h^2-b^2}$

On further simplification

  $=\sqrt{{17}^2-8^2}$

Putting the values in the expression.

  $=\sqrt{289-64}$

After this,

  $=\sqrt{225}$

Taking the square root.

  $p=15$ feet.

Calculation of area of the figure, that is a trapezoid.

  $t=\frac{a+b}{2}\times p$

Putting the values in the equation.

  $=\frac{18+8}{2}\times 15$

On further simplification.

  $=\frac{26}{2}\times 15$

Multiplying

  $=13\times 15$

Total area

  $=195$ square feet.

Area of the unshaded triangle.

  $s=\frac{1}{2}\times b\times p$

Formula of area of the triangle.

  $=\frac{1}{2}\times 8\times 15$

Putting values in the formula.

  $=4\times 15$

Multiply the same.

  $=60$ square feet.

Area of shaded region

  $=t-s$

Subtract area of unshaded region from area of complete figure.

  $=195-60$

Subtracting the same.

  $=135$ square feet.