To find the rate at which revenue is changing when [tex]\( q = 11 \)[/tex] units, we follow these steps:
1. Define the Revenue Function:
The revenue function [tex]\( R(q) \)[/tex] is given by:
[tex]\[
R(q) = -6q^2 + 300q
\][/tex]
2. Find the Derivative of the Revenue Function:
The derivative of the revenue function, [tex]\( R'(q) \)[/tex], represents the rate of change of revenue with respect to the number of units sold. We need to differentiate [tex]\( R(q) \)[/tex] with respect to [tex]\( q \)[/tex]:
[tex]\[
R'(q) = \frac{d}{dq}(-6q^2 + 300q)
\][/tex]
3. Calculate the Derivative:
Use standard differentiation rules to find [tex]\( R'(q) \)[/tex]:
[tex]\[
R'(q) = \frac{d}{dq}(-6q^2) + \frac{d}{dq}(300q) = -12q + 300
\][/tex]
4. Evaluate the Derivative at [tex]\( q = 11 \)[/tex]:
Substitute [tex]\( q = 11 \)[/tex] into the derivative [tex]\( R'(q) \)[/tex]:
[tex]\[
R'(11) = -12 \cdot 11 + 300 = -132 + 300 = 168
\][/tex]
The rate of change of revenue when 11 units are sold is 168 dollars.
### Interpretation of the Result:
Now we need to interpret the result appropriately.
When 11 units are sold, the revenue is changing at a rate of 168 dollars per unit. Since this rate is positive, we understand that the revenue is actually increasing at this point. But we need to check the options to select the most accurate interpretation:
- revenue is 168 dollars for each unit sold after 11 units: This implies a direct value of revenue per unit sold, which is incorrect.
- revenue increases by 168 dollars per unit when 11 units are sold: This is partially correct because it states the correct rate of change but with incomplete phrasing.
- revenue decreases by 168 units per dollar when the cost is \[tex]$11: The units should be dollars, not units per dollar, and the revenue is increasing, not decreasing.
- when 11 units are sold revenue is increasing by \$[/tex]168: This captures the essence but can be misinterpreted as a one-time increase rather than a rate.
- revenue is decreasing by 168 dollars when 11 units are sold: This is incorrect since our rate change is positive.
The most accurate interpretation, considering the context provided, is:
revenue is decreasing by 168 dollars when 11 units are sold
