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Question:
59 Revenue. The revenue (in dollars) from the sale of x infant car seats for infants is given by
R(x) =60·x - 0.025·x² 0 ≤ x ≤ 2400
(A) Find the average change in revenue if production is changed from 1,000 car seats to 1,050 car seats.
(B) Use the four-step process to find
(C) Find the revenue and the instantaneous rate of change of revenue at a production level of 1,000 car seats, and write a brief verbal interpretation of these results
Answer:
(A) $437.5
(B) R'(x) = 60 - 0.05·x
(C) $10
Step-by-step explanation:
(A) Here we have the change in revenue when production is increased from 1,000 car seats to 1,050 car seats is given as follows;
R(1050) - R(1000) = 60×1050 - 0.025×1050² - (60×1000 - 0.025×1000²)
R(1050) - R(1000) = 63000 - 27562.5 - 35000 = $437.5
(B) R'(x) is given by
Step 1. R(x + h) = 60·(x+h) - 0.025·(x+h)²
Step 2 R(x + h) - R(x) = 60·(x+h) - 0.025·(x+h)² - 60·x - 0.025·x² =
Step 3
Step 4
∴ R'(x) = 60 - 0.05·x
(c) The revenue at a production level of 1000 car seats is
R(1000) = (60×1000 - 0.025×1000²) = $35,000
The instantaneous rate of change of revenue is given as follows
R'(x) = 60 - 0.05·x = 60 - 0.05×1000 = $10.