find 2 numbers whose sum is 50 and such that the product of one of them and 5 more than the other is a maximum

Answer 1

You're given x + y = 50 and told to maximize x*(y + 5). y = 50 - x, so you want to maximize x*(55 - x) = 55x - x^2. So take the derivative of that expression: 55 - 2x, and set it equal to zero. If that's zero, then 2x must equal 55, so x = 27.5. y must then equal 50 - 27.5 = 22.5. The numbers are 27.5 and 22.5.

As an aside, note that this makes y + 5 = x. Generally, when given a problem like this, if you think of the sum as being similar to the perimeter of a rectangle and the product similar to the area, the solution which is equivalent to a square is the one that maximizes the area (product).

Answer 2

x + y = 50
x(y + 5) = Max

x + y = 50
y = -x + 50

x(-x + 50 + 5) = Max
x(-x + 55) = Max
-x^2 + 55x = Max
x^2 - 55x = Max

x = (-b)/(2a)

x = (-(-55))/(2(1))
x = (55/2)

x + y = 50
(55/2) + y = 50
55 + 2y = 100
2y = 45
y = (45/2)

ANS : x = 27.5 and y = 22.5

Answer 3

a) x+y=50
b) maximize x*(y+5)

substituting (a) into (b): maximize x*(50-x+5)


to maximize 55x-x^2:
d/dx(55x-x^2) = 0
55 - 2x = 0
x = 27.5

Your pair of numbers is (27.5, 22.5)