Math  /  Calculus

QuestionCurrent Attempt in Progress Let C(q)C(q) represent the cost, R(q)R(q) the revenue, and π(q)\pi(q) the total profit, in dollars, of producing qq items. (a) If C(50)=77C^{\prime}(50)=77 and R(50)=83R^{\prime}(50)=83, approximately how much profit is earned by the 51st 51^{\text {st }} item?
The profit earned from the 51st 51^{\text {st }} item will be approximately $\$ i \square (b) If C(90)=73C^{\prime}(90)=73 and R(90)=69R^{\prime}(90)=69, approximately how much profit is earned by the 91st 91^{\text {st }} item?
The profit earned from the 91st 91^{\text {st }} item will be approximately \ \squarei.(c)If i . (c) If \pi(q)isamaximumwhen is a maximum when q=78,howdoyouthink, how do you think C^{\prime}(78)and and R^{\prime}(78)compare? compare? C^{\prime}(78) \square R^{\prime}(78)$ eTextbook and Media

Studdy Solution

STEP 1

1. The cost function is C(q) C(q) , the revenue function is R(q) R(q) , and the profit function is π(q) \pi(q) .
2. The profit function is defined as π(q)=R(q)C(q) \pi(q) = R(q) - C(q) .
3. The derivative of a function represents the rate of change or marginal value.

STEP 2

1. Calculate the profit from the 51st item using derivatives.
2. Calculate the profit from the 91st item using derivatives.
3. Analyze the relationship between derivatives at maximum profit.

STEP 3

To find the profit from the 51st item, use the derivative of the profit function: π(q)=R(q)C(q) \pi'(q) = R'(q) - C'(q)
Given C(50)=77 C'(50) = 77 and R(50)=83 R'(50) = 83 , calculate: π(50)=R(50)C(50)=8377=6 \pi'(50) = R'(50) - C'(50) = 83 - 77 = 6
The profit from the 51st item is approximately $6 \$6 .

STEP 4

To find the profit from the 91st item, use the derivative of the profit function: π(q)=R(q)C(q) \pi'(q) = R'(q) - C'(q)
Given C(90)=73 C'(90) = 73 and R(90)=69 R'(90) = 69 , calculate: π(90)=R(90)C(90)=6973=4 \pi'(90) = R'(90) - C'(90) = 69 - 73 = -4
The profit from the 91st item is approximately $4-\$4.

STEP 5

If π(q) \pi(q) is a maximum at q=78 q = 78 , then π(78)=0 \pi'(78) = 0 .
This implies: R(78)C(78)=0 R'(78) - C'(78) = 0 R(78)=C(78) R'(78) = C'(78)
Thus, C(78) C'(78) is equal to R(78) R'(78) .
The answers are: (a) \$6 (b) \(-\$4\) (c) \( C'(78) = R'(78) \)

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord