9.2.2 Effects of Changing Dimensions Proportionally (continued) Example 2 Effects of Changing Dimensions Proportionally
The effect of multiplying both the length and width of a rectangle by 3 on the perimeter and area of the rectangle is determined in this example. The perimeter of a rectangle with length 7 cm and width 2 cm is P = 2 + 2 + 7 + 7 = 18 cm. The area of the rectangle is A = (2)(7) = 14 cm 2 . The perimeter of a rectangle with length 21 cm and width 6 cm is P = 6 + 6 + 21 + 21 = 54 cm. The area of the rectangle is A = (6)(21) = 126 cm 2 . Tripling the length and width of the rectangle also triples the perimeter. But tripling the length and width (increasing the dimensions proportionally) increases the area by 3 2 , or 9, times. The effect of multiplying the radius of a circle by 1/5 on the circumference and area of the circle is determined in this example. The circumference of a circle with radius 10 m is C = 2 πr = 20 π m. The area of the circle is A = πr 2 = 100 π m 2 . The circumference of a circle with radius multiplied by 1/5 is C = 2 πr = 2 π (1/5)(10) = 4 π m. The area of the circle is A = πr 2 = π [(1/5)(10)] 2 = 4 π m 2 . The circumference has changed by a factor of 1/5 from 20 π to 4 π . The area has changed by a factor of (1/5) 2 = (1/25) from 100 π to 4 π .
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