R(x)= 360x - 0.04x^2
R(4500) = 810,000
Rewrite: -o.o4x^2+360x
-0.04=a 360=b
Use the vertex equation: v= -b/2a
plug in: -360/2(-0.04) and solve
this is your x, so x=4,500
plug in this number into your original equation and replace it as the x:
-0.04(4,500)^2+360(4,500)
Solve:
-810,000+160,000= 810,000
x=810,000
maximum revenue = 810,000
The equation R(x) = 360x - 0.04x^2 represents a quadratic function, where R(x) is the revenue (in dollars) generated by selling x items. The equation is in the form of a quadratic polynomial, with the term -0.04x^2 representing the negative coefficient of x^2, indicating that the revenue function is a downward-opening parabola.
To find the maximum value of the revenue function, we can use calculus. By taking the derivative of R(x) with respect to x and setting it equal to zero, we can find the critical points of the function.
Let's find the derivative of R(x):
R'(x) = 360 - 0.08x
Setting R'(x) equal to zero:
360 - 0.08x = 0
Next, solve for x:
0.08x = 360
x = 360 / 0.08
x = 4500
Therefore, the critical point is x = 4500.
To determine if this critical point is a maximum or minimum, we can take the second derivative of R(x) and evaluate it at the critical point.
Let's find the second derivative, R''(x):
R''(x) = -0.08
Since the second derivative is a constant (-0.08), and it is negative, this means that the revenue function R(x) is concave down, indicating a maximum.
Hence, the maximum value of the revenue function R(x) = 360x - 0.04x^2 is attained when x = 4500, and the maximum revenue can be found by substituting this value back into the equation.