The profit for a company that sells chairs is defined by P= -5x^2+200x+200. Where P is profit and x is the number of chairs produced. Find the number of chairs "x" that will produce the maximum profit "P".
Since the title is "algebra", I assume calculus cannot be used.
In this case, complete squares of P(x).
P=-5(x-20)²+2200
which means that P(x) is at a maximum when x=20, and P(x)=2200.
To find the number of chairs that will produce the maximum profit, we need to determine the value of "x" that corresponds to the maximum point on the profit function. In this case, the profit function is defined by the equation P = -5x^2 + 200x + 200.
To find the maximum point, we can use calculus. We start by finding the derivative of the profit function with respect to "x" and setting it equal to zero to find critical points. Let's differentiate the function:
dP/dx = -10x + 200
Now, we set the derivative equal to zero and solve for "x":
-10x + 200 = 0
-10x = -200
x = -200 / -10
x = 20
So, we have found that x = 20 is a critical point of the profit function.
To confirm if this point is a maximum or minimum, we can take the second derivative of the profit function. If the second derivative is positive, then the critical point is a minimum; if it is negative, the critical point is a maximum.
Let's find the second derivative by differentiating the first derivative:
d^2P/dx^2 = -10
The second derivative is a constant value of -10, which is negative. This indicates that the critical point x = 20 corresponds to a maximum profit.
Therefore, the number of chairs "x" that will produce the maximum profit "P" is 20.