Let C(q) represent the cost, R(q) the revenue, and π(q) the total profit, in dollars, of producing q items.
a) If C′(67) = 65 and R′(67) = 86, approximately how much profit is earned by item number 68?
b) If C′(149) = 120 and R′(149) = 98, approximately how much profit is earned by item number 150?
marginal profit = R'(x)-C'(x)
a).
Profit for item 68
= π'(67)
= R'(67)-C'(67)
= 86-65
= $21
I'll let you try b).
To find the approximate profit earned by item number 68, we need to use the provided derivative values and apply them to the profit function.
a) Given:
C'(67) = 65
R'(67) = 86
We can use the derivative of the profit function, π'(q), to calculate the profit earned by item number 68.
π'(q) = R'(q) - C'(q)
Since we know the derivative values at q = 67, we can substitute them into the formula and solve for π'(67):
π'(67) = R'(67) - C'(67)
= 86 - 65
= 21
This means that for each item produced beyond item number 67, the profit increases by approximately $21.
To find the profit earned by item number 68, we can add the calculated derivative value to the total profit earned up to item number 67.
π(68) = π(67) + π'(67)
We are not given the specific values of π(67), so we cannot calculate the exact profit. However, we can still approximate it based on the derivative value. Let's assume that π(67) is $100.
π(68) ≈ 100 + 21
= 121
Therefore, the approximate profit earned by item number 68 is $121.
b) Given:
C'(149) = 120
R'(149) = 98
Using a similar approach as in part a), we can calculate the approximate profit earned by item number 150.
π'(149) = R'(149) - C'(149)
= 98 - 120
= -22
Notice that in this case, the derivative value is negative. This means that producing each additional item beyond item number 149 actually results in a loss of profit.
Assuming that π(149) is $500, we can apply the derivative value to calculate the approximate profit earned by item number 150.
π(150) ≈ π(149) + π'(149)
= 500 - 22
= 478
Therefore, the approximate profit earned by item number 150 is $478, but remember that this result indicates a loss compared to producing only 149 items.
To determine the approximate profit earned by item number 68, we can use the information given about the derivative of the cost and revenue functions at item number 67.
a) First, let's start by understanding what C'(67) = 65 and R'(67) = 86 mean. The derivative C'(67) = 65 represents the rate of change of the cost function at item number 67, which indicates that the cost is increasing by 65 dollars per item. Similarly, R'(67) = 86 represents the rate of change of the revenue function at item number 67, indicating that the revenue is increasing by 86 dollars per item.
To find the profit earned by item number 68, we can use the formula: π(q) = R(q) - C(q).
Since we have information about the derivative of the cost and revenue functions at item number 67, we can use it to estimate the cost and revenue at item number 68.
Using the derivative C'(67) = 65, we can estimate the cost at item number 68 as follows: C(68) = C(67) + C'(67) * (68 - 67). Therefore, C(68) = C(67) + 65 * 1.
Similarly, using the derivative R'(67) = 86, we can estimate the revenue at item number 68 as follows: R(68) = R(67) + R'(67) * (68 - 67). Therefore, R(68) = R(67) + 86 * 1.
Now that we have estimates for the cost and revenue at item number 68, we can find the profit as follows: π(68) = R(68) - C(68).
b) Using the same approach, we can approximate the profit earned by item number 150.
First, we have C'(149) = 120 and R'(149) = 98, representing the rate of change of the cost and revenue functions, respectively, at item number 149.
We can estimate the cost and revenue at item number 150 similarly to the previous example:
C(150) = C(149) + C'(149) * (150 - 149)
R(150) = R(149) + R'(149) * (150 - 149)
Finally, we can find the profit as follows: π(150) = R(150) - C(150).
By following these steps, you should be able to estimate the profit earned by item number 68 and 150 based on the given information.