QuestionMonthly revenue is and cost is . Find wristwatches for max revenue and profit, then explain differences.
Studdy Solution
STEP 1
Assumptions1. The monthly revenue is given by the function , where is the number of wristwatches sold.
. The monthly cost is given by the function , where is the number of wristwatches sold.
3. The profit $$ is given by the function $(x)=R(x)-C(x)$, where $x$ is the number of wristwatches sold.
STEP 2
To find the number of wristwatches that must be sold to maximize revenue, we need to find the maximum of the revenue function . This can be done by finding the derivative of and setting it equal to zero.
STEP 3
Now, set the derivative equal to zero and solve for .
STEP 4
olve the equation for .
STEP 5
Calculate the value of .
Since the number of wristwatches sold must be an integer, we round to the nearest integer. Therefore, the firm must sell198 wristwatches to maximize revenue.
STEP 6
To find the maximum revenue, substitute into the revenue function .
STEP 7
Calculate the maximum revenue.
STEP 8
The profit function is given by . Substitute the given functions and into the profit function.
STEP 9
implify the profit function.
STEP 10
To find the number of wristwatches that must be sold to maximize profit, we need to find the maximum of the profit function . This can be done by finding the derivative of and setting it equal to zero.
STEP 11
Now, set the derivative equal to zero and solve for .
STEP 12
olve the equation for .
STEP 13
Calculate the value of .
Since the number of wristwatches sold must be an integer, we round to the nearest integer. Therefore, the firm must sell133 wristwatches to maximize profit.
STEP 14
To find the maximum profit, substitute into the profit function .
STEP 15
Calculate the maximum profit.
STEP 16
The answers found in parts (a) and (c) differ because the number of wristwatches that maximizes revenue is not the same as the number of wristwatches that maximizes profit. This is due to the fact that the cost of producing and selling the wristwatches is not taken into account when calculating the revenue, but it is when calculating the profit.
A quadratic function is a reasonable model for revenue because it allows for the possibility of a maximum revenue. The revenue increases with the number of wristwatches sold up to a certain point, after which it starts to decrease. This is due to the fact that the price per wristwatch decreases as more wristwatches are sold.
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