Suppose the cost function for producing gadgets is C(q)=0.05q2+3q+48.
a. At what production level is the maximal profit achieved if each item is sold for 9 dollars? ______ items.
b. What is the maximum profit if each item is sold for 9 dollars? ______ dollars.
To find the production level that maximizes profit and the corresponding maximum profit, we need to determine the quantity that maximizes the difference between revenue and cost.
a. The revenue function can be determined by multiplying the number of items produced (q) by the selling price per item, which is $9 in this case. Therefore, the revenue function is R(q) = 9q.
The profit function is given by:
Profit = Revenue - Cost
Substituting the expressions for revenue and cost, we find:
Profit = R(q) - C(q)
Profit = 9q - (0.05q^2 + 3q + 48)
Profit = 9q - 0.05q^2 - 3q - 48
Rearranging the terms:
Profit = -0.05q^2 + 6q - 48
To find the production level that maximizes profit, we need to find the quantity (q) where the derivative of the profit function equals zero.
Taking the derivative of the profit function with respect to q:
dProfit/dq = -0.1q + 6
Setting the derivative equal to zero and solving for q:
-0.1q + 6 = 0
-0.1q = -6
q = -6 / -0.1
q = 60
Therefore, the production level at which the maximal profit is achieved is 60 items.
b. To find the maximum profit, substitute the value of q into the profit function:
Profit = -0.05q^2 + 6q - 48
Profit = -0.05(60)^2 + 6(60) - 48
Profit = -0.05(3600) + 360 - 48
Profit = -180 + 360 - 48
Profit = 132 dollars
Therefore, the maximum profit when each item is sold for $9 is $132.
To find the production level at which the maximal profit is achieved and the maximum profit itself, we will need to use the given cost function and the selling price. The profit function can be expressed as P(q) = R(q) - C(q), where R(q) represents the revenue at a given production level q.
a. To determine the production level at which the maximal profit is achieved, we need to find the value of q that maximizes the profit function P(q). We can do this by finding the critical points of the profit function. The critical points occur where the derivative of the profit function is equal to zero.
First, let's find R(q), the revenue function. The revenue function is given by multiplying the selling price per item by the production level: R(q) = 9q.
Now, let's substitute the revenue function and the cost function into the profit function: P(q) = R(q) - C(q) = 9q - (0.05q^2 + 3q + 48).
To find the critical points, we need to find where the derivative of the profit function is equal to zero. Let's differentiate the profit function:
P'(q) = 9 - (0.1q + 3).
Setting the derivative equal to zero and solving for q:
9 - (0.1q + 3) = 0
0.1q + 3 = 9
0.1q = 6
q = 60.
Therefore, the production level at which the maximal profit is achieved is 60 items.
b. To find the maximum profit, we substitute the production level (q = 60) into the profit function: P(60) = 9(60) - (0.05(60)^2 + 3(60) + 48).
P(60) = 540 - (0.05(3600) + 180 + 48)
P(60) = 540 - (180 + 180 + 48)
P(60) = 540 - 408
P(60) = 132.
Therefore, the maximum profit achieved if each item is sold for $9 is $132.
revenue = price * quantity, so
r(q) = 9q
profit = revenue - cost, so
p(x) = 9q - 0.05q^2 - 3q - 48 = -0.05q^2 + 6q - 48
dp/dq = -0.1q + 6
max profit is $132 at q=60