SOLUTION: I have a homeowrk question that states: find the maximum revenue for the revenue function R(x)=490x-0.07x^2. How do I work this problem out? I have tried everything.

Algebra ->  Functions -> SOLUTION: I have a homeowrk question that states: find the maximum revenue for the revenue function R(x)=490x-0.07x^2. How do I work this problem out? I have tried everything.      Log On


   

Question 86040: I have a homeowrk question that states: find the maximum revenue for the revenue function R(x)=490x-0.07x^2. How do I work this problem out? I have tried everything.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
R%28x%29=490x-0.07x%5E2

R%28x%29=490x-%287%2F100%29x%5E2 Make into 0.07 into a fraction


R%28x%29=-%287%2F100%29x%5E2%2B490x Rearrange the terms
Now lets convert this to vertex form to find the max revenue
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=%28-7%2F100%29+x%5E2%2B490+x%2B0 Start with the given equation



y-0=%28-7%2F100%29+x%5E2%2B490+x Subtract 0 from both sides



y-0=%28-7%2F100%29%28x%5E2-7000x%29 Factor out the leading coefficient %28-7%2F100%29



Take half of the x coefficient -7000 to get -3500 (ie %281%2F2%29%28-7000%29=-3500).


Now square -3500 to get 12250000 (ie %28-3500%29%5E2=%28-3500%29%28-3500%29=12250000)





y-0=%28-7%2F100%29%28x%5E2-7000x%2B12250000-12250000%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 12250000 does not change the equation




y-0=%28-7%2F100%29%28%28x-3500%29%5E2-12250000%29 Now factor x%5E2-7000x%2B12250000 to get %28x-3500%29%5E2



y-0=%28-7%2F100%29%28x-3500%29%5E2-%28-7%2F100%29%2812250000%29 Distribute



y-0=%28-7%2F100%29%28x-3500%29%5E2%2B857500 Multiply



y=%28-7%2F100%29%28x-3500%29%5E2%2B857500%2B0 Now add 0 to both sides to isolate y



y=%28-7%2F100%29%28x-3500%29%5E2%2B857500 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=-7%2F100, h=3500, and k=857500. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=%28-7%2F100%29x%5E2%2B490x%2B0 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%28-7%2F100%29x%5E2%2B490x%2B0%29 Graph of y=%28-7%2F100%29x%5E2%2B490x%2B0. Notice how the vertex is (3500,857500).



Notice if we graph the final equation y=%28-7%2F100%29%28x-3500%29%5E2%2B857500 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%28-7%2F100%29%28x-3500%29%5E2%2B857500%29 Graph of y=%28-7%2F100%29%28x-3500%29%5E2%2B857500. Notice how the vertex is also (3500,857500).



So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.






So the maximum revenue that can be generated is $857,500 (the y-coordinate of the vertex) by selling 3,500 units (the x-coordinate of the vertex)