QuestionMonthly revenue from selling wristwatches is and cost is . Find max revenue, profit function, and max profit. Explain differences in results.
Studdy Solution
STEP 1
Assumptions1. The monthly revenue is given by the function
. The monthly cost is given by the function
3. The profit $$ is given by the function $(x)=R(x)-C(x)$4. We are looking for the number of wristwatches to sell to maximize revenue and profit, and the maximum revenue and profit.
STEP 2
To find the number of wristwatches to sell to maximize revenue, we need to find the maximum point of the revenue function . This is done by taking the derivative of the function and setting it equal to zero.
STEP 3
Set the derivative equal to zero and solve for .
STEP 4
olve for .
STEP 5
Calculate the value of .
STEP 6
Since we can't sell half a wristwatch, we need to round to the nearest integer. So, the firm must sell198 wristwatches to maximize revenue.
STEP 7
To find the maximum revenue, plug into the revenue function .
STEP 8
Calculate the maximum revenue.
STEP 9
The profit function is given by . So, we just need to substitute and into this equation.
STEP 10
implify the profit function.
STEP 11
To find the number of wristwatches to sell to maximize profit, we need to find the maximum point of the profit function . This is done by taking the derivative of the function and setting it equal to zero.
STEP 12
Set the derivative equal to zero and solve for .
STEP 13
olve for .
STEP 14
Calculate the value of .
STEP 15
Since we can't sell half a wristwatch, we need to round to the nearest integer. So, the firm must sell133 wristwatches to maximize profit.
STEP 16
To find the maximum profit, plug into the profit function .
STEP 17
Calculate the maximum profit.
STEP 18
The answers found in parts (a) and (c) differ because the revenue and cost functions are different. The revenue function is a quadratic function, which means it has a maximum value. The cost function is a linear function, which means it increases constantly. Therefore, the number of wristwatches to sell to maximize revenue and profit are different.
A quadratic function is a reasonable model for revenue because it reflects the reality of business operations. At the beginning, as the number of wristwatches sold increases, the revenue increases. However, after reaching a certain point, selling more wristwatches may lead to a decrease in revenue due to factors such as market saturation and increased competition.
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