QuestionMonthly revenue from selling wristwatches is and cost is .
(a) Find for max revenue and max revenue amount.
(b) Profit function is .
(c) Find for max profit and max profit amount.
(d) Explain differences in (a) and (c) and why a quadratic model works for revenue.
Studdy Solution
STEP 1
Assumptions1. The revenue function is given by
. The cost function is given by
3. The profit function is given by 4. The number of wristwatches sold to maximize revenue is the value of that maximizes 5. The number of wristwatches sold to maximize profit is the value of that maximizes
STEP 2
To find the number of wristwatches that must be sold to maximize revenue, we need to find the maximum of the function . This can be done by finding the derivative of and setting it equal to zero.
STEP 3
Calculate the derivative of .
STEP 4
Set the derivative equal to zero and solve for .
STEP 5
olve the equation for .
STEP 6
Calculate the value of .
STEP 7
Since the number of wristwatches sold cannot be a fraction, we round to the nearest integer.
STEP 8
Now that we have the number of wristwatches that must be sold to maximize revenue, we can find the maximum revenue by plugging into .
STEP 9
Calculate the maximum revenue.
STEP 10
The profit function is given by . Substitute the given functions and into the equation.
STEP 11
implify the profit function.
STEP 12
To find the number of wristwatches that must be sold to maximize profit, we need to find the maximum of the function . This can be done by finding the derivative of and setting it equal to zero.
STEP 13
Calculate the derivative of .
STEP 14
Set the derivative equal to zero and solve for .
STEP 15
olve the equation for .
STEP 16
Calculate the value of .
STEP 17
Since the number of wristwatches sold cannot be a fraction, we round to the nearest integer.
STEP 18
Now that we have the number of wristwatches that must be sold to maximize profit, we can find the maximum profit by plugging into .
STEP 19
Calculate the maximum profit.
STEP 20
The answers found in parts (a) and (c) differ because the number of wristwatches that maximizes revenue is not the same as the number that maximizes profit. This is because the cost of producing and selling the wristwatches also has to be taken into account when calculating profit.A quadratic function is a reasonable model for revenue because it allows for the fact that revenue will increase with the number of wristwatches sold up to a certain point, but then decrease as the number of wristwatches sold continues to increase. This is due to the law of diminishing returns, which states that after a certain point, each additional unit of a product sold will generate less revenue than the previous unit.
Was this helpful?