What is the maximum product of two numbers that add to 18? What numbers yield th
ID: 3411236 • Letter: W
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18. Let the two numbers be x and y.Then x + y = 26 so that y = 26 - x. Then the product P = xy = x ( 26 -x) = -x2 + 26x = - (x2 -26x) = - (x2 - 26x + 169) + 169 = -( x - 13)2 + 169. This is a parabola opening downwards with vertex at ( 13, 169). Since the maximum of a parabola opening downwards occurs at the vertex, therefore, P is maximum when x = 13. Then y = 26 - 13 = 13 and the maximum of P is x*y = 13*13 = 169.The numbers that yield this product are 13, 13.
19. Let the two numbers be x and y and let x be the larges of the two numbers. Then x - y = 8 so that y = x - 8.Then the product P = xy = x ( x - 8) = x2 -8x = (x2 -8x + 16 ) - 16 = ( x - 4)2 - 16. This is a parabola opening upwards with vertex at ( 4, -16). Since the minimum of a parabola opening upwards occurs at the vertex, therefore, P is minimum when x = 4. Then y = x - 4 = 4 - 8 = - 4 and the minimum of P is x*y = 4* (-4 )= -16 The numbers that yield this product are 4, - 4.
20.Let the two numbers be x and y and let x be the larges of the two numbers. Then x - y = 7 so that y = x - 7.Then the product P = xy = x ( x - 7) = x2 -7x = (x2 -2*7/2 x + 49/4 ) - 49/4 = ( x - 7/2)2 - 49/4. This is a parabola opening upwards with vertex at ( 7/2, -49/4). Since the minimum of a parabola opening upwards occurs at the vertex, therefore, P is minimum when x = 7/2. Then y = x - 7 = 7/2-7 = -7/2 and the minimum of P is x*y = 7/2*( -7/2) = -49/4 The numbers that yield this product are 7/2, - 7/2.
21. Let the two numbers be x and y. Then x + y = -10 so that y = -10 - x. Then the product P = xy = x ( -10 -x) = - x2 -10x = - (x2 + 10x) = - (x2 + 10x + 25) + 25 = -( x + 5)2 + 25. This is a parabola opening downwards with vertex at ( -5, 25). Since the maximum of a parabola opening downwards occurs at the vertex, therefore, P is maximum when x = -5. Then y = -10 - (-5) = -5 and the maximum of P is x*y = (-5)*(-5) = 25.The numbers that yield this product are -5, -5 .
22. Let the two numbers be x and y. Then x + y = -12 so that y = -12 - x. Then the product P = xy = x ( -12 -x) = - x2 -12x = - (x2 + 12x) = - (x2 + 12x + 36) + 36 = -( x + 6)2 + 36. This is a parabola opening downwards with vertex at ( -6, 36). Since the maximum of a parabola opening downwards occurs at the vertex, therefore, P is maximum when x = -6. Then y = -12 - (-6) = -6 and the maximum of P is x*y = (-6)*(-6) = 36.The numbers that yield this product are -6, -6 .
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