To determine the rate at which revenue changes when 165 units are sold, follow these steps:
1. Identify the Revenue Function:
The revenue function is [tex]\( R(q) = -2q^2 + 500q \)[/tex], where [tex]\( R(q) \)[/tex] is the revenue in dollars and [tex]\( q \)[/tex] is the number of units sold.
2. Find the First Derivative of the Revenue Function:
The first derivative of [tex]\( R(q) \)[/tex] with respect to [tex]\( q \)[/tex], denoted [tex]\( R'(q) \)[/tex], represents the rate of change of revenue with respect to the number of units sold. It is given by:
[tex]\[
R'(q) = \frac{d}{dq}(-2q^2 + 500q)
\][/tex]
Calculating the derivative, we get:
[tex]\[
R'(q) = -4q + 500
\][/tex]
3. Evaluate the Derivative at [tex]\( q = 165 \)[/tex]:
To find the rate of change of revenue when 165 units are sold, substitute [tex]\( q = 165 \)[/tex] into [tex]\( R'(q) \)[/tex]:
[tex]\[
R'(165) = -4(165) + 500
\][/tex]
Performing the arithmetic:
[tex]\[
R'(165) = -660 + 500 = -160
\][/tex]
4. Interpret the Result:
The value [tex]\( R'(165) = -160 \)[/tex] means that the revenue is decreasing at a rate of 160 dollars per unit when 165 units are sold.
Thus, the correct interpretation of the result is:
- revenue is decreasing by 160 dollars per unit when 165 units are sold
