Question

calcula el área de la región sombreada
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Answer 1

PARA LA FIGURA NUMERO 35.-

Son dos cuadrados diferentes y son dos rectangulos iguales, de aqui =

LA FORMA SENCILLA =

Sea el area de todo el cuadrado = "k"

$(5a + b )( 5a + b) = k \\ (5a)(5a + b) + (b)(5a + b) = k \\( 25 {a}^{2} + 5ab) + (5ab + {b}^{2} ) = k \\ 25 {a}^{2} + 2(5ab) + {b}^{2} = k \\ 25 {a}^{2} + 10ab + {b}^{2} = k$

CALCULAMOS SUS PARTES DE COLOR BLANCO =

Para el cuadrado de color Blanco sea su area = "m"

$(5a - b)(5a - b) = m \\ (5a)(5a - b) + ( - b)(5a - b) = m \\( 25 {a}^{2} - 5ab) + (- 5ab + {b}^{2} ) = m \\ 25 {a}^{2} - 2(5ab) + {b}^{2} = m \\ 25 {a}^{2} - 10ab + {b}^{2} = m$

Para los rectangulos iguales, solamente calculamos uno y el valor lo multiplicamos por dos, y su area sea = "t"

$</em></strong></p><p><strong><em>[tex]\frac{t}{2} = ( 5a - b)(b) \\ \frac{t}{2} = (5ab - {b}^{2} ) \\ t = (5ab - {b}^{2} ) \times 2 \\ t = 10ab - {2b}^{2}$

EL AREA ENTONCES ES =

$a = k - ( m + 2t) \\ a = (25 {a}^{2} + 10ab + {b}^{2} ) - (25 {a}^{2} - 10ab + {b}^{2} + 10ab - 2 {b}^{2} ) \\ a = (25 {a}^{2} + 10ab + {b}^{2} ) - (25 {a}^{2} ( - 10ab + 10ab) + ( {b}^{2} - 2 {b}^{2} )) \\ a = (25 {a}^{2} + 10ab + {b}^{2} ) - (25 {a}^{2} - {b}^{2} ) \\ a = (25 {a - 25 {a}^{2} })+ {(b - b)}^{2} + 10ab \\ a = 10ab$

PARA LA FIGURA 36.-

Son dos cuadrado los propuestos, asi que =

AREA DEL CUADRADO GRANDE = "z"

$z = (6x + 5y - 3)(6x + 5y - 3) \\ z = (6x)(6x + 5y - 3) + (5y)(6x + 5y - 3) + ( - 3)(6x + 5y - 3) \\ z = (36 {x}^{2} + 30xy - 18x) + (30xy + 25 {x}^{2} - 15y) + ( - 18x - 15y + 9) \\ z = (36 {x}^{2} + 25 {y}^{2} + 9) + 2(30xy) - 2(15y) - 2(18x) \\ z =( 36 {x}^{2} + 25 {y}^{2} + 9 + 60xy - 30y - 36x)$

AREA DEL CUADRADO CHICO = "o"

$o = (7x - 8y)(7x - 8y )\\ o = (7x)( 7x- 8y) + ( - 8y)(7x - 8y) \\ o = (49 {x}^{2} - 56xy) + ( - 56xy + 64 {y}^{2} ) \\ o = ( {49 {x} }^{2} + 64 {y}^{2} ) - 2(56xy) \\ o =( 49 {x}^{2} + 64 {y}^{2} + 122xy)$

AHORA SEA EL AREA = "a"

$a = z - o \\ a =( 36 {x}^{2} + 25 {y}^{2} + 9 + 60xy - 30y - 36x) - (49 {x}^{2} - 64 {y}^{2} - 122xy) \\ a = (36 {x}^{2} - 49 {x}^{2} ) + (25 {y}^{2} - 64 {y}^{2} ) + (60xy - 122xy) (- 30y - 36x + 9) \\ a = ( - 13 {x}^{2} + 39 {y}^{2} - 62xy - 30y - 36x + 9)$

SALUDOSSSSS!!!!....