Final Answer:
The revenue is increasing at a rate of $28,600 per day when 170 units have been sold, This rate of change does not match any of the options provided .
Step-by-step explanation:
To find how rapidly revenue is increasing, we need to determine the rate of change of the revenue function with respect to time.
This involves the concept of related rates, which is a part of calculus, specifically derivative application.
Firstly, the provided revenue function is:
R(x) = 1200x - 2x^2
We need to find the derivative of the revenue with respect to the number of units, x, because the rate of change of revenue is dependent on the rate of change of the number of units sold. This is known as the marginal revenue.
So, we find the first derivative of R with respect to x:
R'(x) =d/dx(1200x - 2x^2)
R'(x) = 1200 - 4x
Now, we are given that sales are increasing at the rate of 55 units per day. This is written as:
dx/dt = 55
where dx/dt is the rate of change of x with respect to time t.
We are asked to find how rapidly revenue is increasing when 170 units have been sold, so we need to use the derivative R'(x) at x = 170.
Substitute x = 170 into R'(x):
R'(170) = 1200 - 4(170)
R'(170) = 1200 - 680
R'(170) = 520
This means that for each additional unit sold, revenue increases by $520. However, we need to find how much revenue increases for each day, given that 55 units are sold per day.
Thus, we multiply the rate of change of revenue with respect to x by the rate of change of x with respect to time:
Rate of change of revenue per day = R'(x)dx/dt
Rate of change of revenue per day = 520 × 55
Rate of change of revenue per day = 28600
Therefore, the revenue is increasing at a rate of $28,600 per day when 170 units have been sold.
This rate of change does not match any of the options provided (A) $2,500 per day, (B) $3,750 per day, (C) $4,250 per day, (D) $6,000 per day.
However, the answer based on the mathematical procedure is $28,600 per day.