SOLUTION: A machine comes in 2 sections. The weights of the section are x kg and b kg. The cost, c, of the machine (in dollars) is given by c=2x+b. The earning capacity, y, of the machine is

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Question 1107689: A machine comes in 2 sections. The weights of the section are x kg and b kg. The cost, c, of the machine (in dollars) 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. [Please do not use differentiation].
Answer by Theo(13342) About Me  (Show Source):
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you have 2 equations.

c = 2x + b is the first equation.

y = x * (x + b) is the second equation.

you are given that c = 10 which makes the first equation 10 = 2x + b.

if you solve this first equation for b, you get b = 10 - 2x.

replace b in the second equation with this to get y = x * (x + 10 - 2x).

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

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

reorder the terms in descending order of degree to get y = -x^2 + 10x.

that's a quadratic equation in the form of y = ax^2 + bx + c.

in that equation, a = -1 and b= 10.

the maximum value of y in that equation is when x = -b/2a.

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

when x = 5, the value of y is equal to -5^2 + 10*5= -25 + 50 = 25.

your maximum value of y should be 25.

if you graph the equation of y = -x^2 + 10x, it will look like this.

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you can see that the maximum value of the equation is y = 25 when x = 5.