The revenue function for a bicycle shop is given by R(x) = x · p(x)
R
x
x
p
x
dollars where x
x
is the number of units sold and p(x) = 100 − 0.4x
p
x
100
0
4
x
is the unit price. Find the maximum revenue.
how about fixing the formatting?
To find the maximum revenue, we need to determine the value of x that maximizes the revenue function R(x) = x * p(x).
First, we need to find the derivative of the revenue function R(x) with respect to x. The derivative will help us determine the value of x where the rate of change of revenue is zero, which corresponds to the maximum revenue.
Let's find the derivative of R(x):
R(x) = x * p(x)
R(x) = x * (100 - 0.4x)
R(x) = 100x - 0.4x^2
To find the derivative of R(x), we apply the power rule and constant multiple rule:
R'(x) = 100 - 0.8x
Now, let's set the derivative equal to zero and solve for x to find the critical point(s):
R'(x) = 100 - 0.8x = 0
0.8x = 100
x = 100 / 0.8
x = 125
Since we have only one critical point, we can plug this value back into the revenue function to find the maximum revenue:
R(x) = x * p(x)
R(125) = 125 * (100 - 0.4(125))
R(125) = 125 * (100 - 50)
R(125) = 125 * 50
R(125) = 6250
Therefore, the maximum revenue for the bicycle shop is $6250.