Find two numbers whose difference is 168 and whose product is a minimum.?

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5 Answers

  • If two numbers add up to a certain amount then the biggest product is always when the two numbers are the same.

    In this case …

    Half168 is 84.

    So the numbers are 84 and -84

    because the product is MINUS 7056 !

  • Rewrite using math symbols.

    Assuming largest number is x

    x – y = 168 ==> y = x – 168

    xy = P where P is a minimum

    x(x-168) = P. This is a parabola.

    We can find vertex (minimum) by putting in vertex format.

    Use completing square technique

    x2 – 168x = P

    x^2 – 168x + 7056 = P + 7056 (add 7056 to complete square)

    (x – 84)^2 = P + 7056.

    Vertex occurs where x = 84. y = 84 -168 = -84.

    Answer 84, -84.

  • But if you are not in calculus:

    x = one number

    y = the other (higher)

    y – x = 168

    y = 168 + x

    product = xy = x(168 + x) = 168x + x^2 = 1x^2 + 168x

    If y = ax^2 + bx + c and a is positive,

    a minimum occurs when x = -b / (2a)

    so x = -(168) / (2 • 1) = -168/ 2 = -84

    and y = 168 + x = 168 + (-84) = 84

  • y = x-168

    p = product = xy = x^2-168x

    dP/dx = 2x -168 = 0

    x = 84

    y = -84

    Numbers are -84 and 84

  • x – y = 168

    P = x * y

    P = x * y

    P = (168 + y) * y

    P = 168 * y + y^2

    dP/dy = 168 + 2y

    dP/dy = 0

    0 = 168 + 2y

    0 = 84 + y

    y = -84

    x – y = 168

    x – (-84) = 168

    x + 84 = 168

    x = 84

    84 and -84

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