Question 1012464
A machine comes in 2 parts, which weigh 'x' kg and 'b' kg respectively. The cost 'c' of the machine is given by c = 2x + b. The earning capacity 'y' of the machine is given by y = x(x+b). If 'c' has the fixed value 10, express y as a function of x and hence find the value of x for which y is a maximum. Find the maximum value of y.


c = cost = 2x + b.


since c = cost = 10, you get 10 = 2x + b.


solve for b to get b = 10 - 2x.


y = x * (x+b)


since b = 10 - 2x, you get y = x * (x + 10 - 2x)


simplify to get y = x * (10 - x)


simplify further to get y = 10x - x^2


set this equation into standard quadratic form of 0 = ax^2 + bx + o get y = -x^2 + 10x.


this means that:
a = -1
b = 10
c = 0


the x value of the max/min point of a quadratic equation is at x = -b/2a.


this becomes x = -10/-2 which results in x = 5.


when x = 5, y = -x^2 + 10x becomes y = -25 + 50 which becomes y = 25.


the maximum earning capacity is therefore 25 units of whatever denomination you are using.


you can graph this equation and it will show you the same result visually.


look below:


<img src = "http://theo.x10hosting.com/2016/010603.jpg" alt="$$$" </>