9.1.3 Composite Figures (continued)
Example 3 Engineering Application
The area of a composite figure is determined by addition and used to calculate energy requirements for a room in this application example. Various dimensions of the room are given. The base lengths and side length of the trapezoid are given. To calculate the height of the trapezoid, recognize that the height line forms a right triangle on the end with side lengths in a 3, 4, 5 Pythagorean triple. Substituting the base lengths and height into the formula for the area of a trapezoid gives an area of 60 ft 2 . The area of the rectangle is the product of the base and the height, or 360 ft 2 . The length of the radius of the semicircle is one-half the width of the rectangle, or 9 ft. Therefore, the area of the semicircle is 81 π /2 ft 2 . The total area of the theater times 20 BTUs per ft 2 is equal to about 10,900 BTUs required to air-condition the theater.
Example 4 Estimating Areas of Irregular Shapes
A composite figure is used to estimate the area of an irregular shape. The length of the side of a square in the grid is given as 1 cm. Begin by drawing figures within the irregular shape in forms that can have their areas calculated. The figure can be approximated by the four triangles and one square shown. The square has sides of 2 cm, so the area of the square is 2 2 = 4 cm 2 . The triangles have base lengths of 2 cm and heights of 3 cm. Their combined area is (4)(1/2)(2)(3) = 12 cm 2 . The total estimate of the area of the irregular shape is 4 + 12 cm 2 = 16 cm 2 .
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