Billy Jeans' profit is given by the equation y=2.7-.0000015(x-2550)^2, where x represents the number of units manufactured and y represents the profit given in millions of dollars. What is the maximum possible profit?
a) 2.55 billion dollars
b) -7.1 million dollars
c) 9.8 million dollars
d) 2.7 million dollars
Calculate dy/dx and set it equal to zero.
dy/dx = 1.5*10^-2*(x-2550)=0
x = 2550 at the maximum
Y (@ x=2550) = 2.7 million
The answer is (d)
I was hoping to using this as my a example could you show me how to get to the answer. I'm horrible at math.
To find the maximum possible profit, we need to calculate dy/dx and set it equal to zero.
Given the profit equation, y = 2.7 - 0.0000015(x - 2550)^2, we need to differentiate it with respect to x to find dy/dx.
First, we can simplify the equation by expanding (x - 2550)^2:
y = 2.7 - 0.0000015(x^2 - 5100x + 2550^2)
= 2.7 - 0.0000015x^2 + 0.00765x - 3.27825
Now, we take the derivative of y with respect to x:
dy/dx = -0.000003x + 0.00765
Next, we set dy/dx equal to zero and solve for x:
-0.000003x + 0.00765 = 0
-0.000003x = -0.00765
x = -0.00765 / (-0.000003)
x = 2550
So, the value of x at the maximum profit is 2550.
To find the maximum profit, we substitute this value of x into the original equation:
y = 2.7 - 0.0000015(2550 - 2550)^2
y = 2.7 - 0.0000015(0)
y = 2.7
Therefore, the maximum possible profit is 2.7 million dollars, which corresponds to option (d).