A company is manufacturing kayaks and can sell all that it manufactures. The revenue (in dollars) is given by R=750-(X^2/30) where the production output in 1 day is x kayaks. If production is increasing at 3 kayaks per day when production is 40 kayaks per day, find the rate of increase in revenue.
2292
To find the rate of increase in revenue, we need to take the derivative of the revenue function with respect to production output (x) and then evaluate it when production is 40 kayaks per day.
First, let's find the derivative of the revenue function. The revenue function is given by:
R = 750 - (x^2/30)
To find the derivative, we differentiate each term separately. The derivative of 750 with respect to x is 0, because it is a constant. For the second term, we can rewrite it as (1/30) * x^2. The derivative of x^2 with respect to x is 2x. So, the derivative of the revenue function is:
dR/dx = 0 - (2x/30)
Now, let's evaluate the derivative when production is 40 kayaks per day:
dR/dx = - (2(40)/30)
= - (80/30)
= -8/3
Therefore, the rate of increase in revenue is -8/3 dollars per kayak per day. This means that for every increase of 1 kayak per day, the revenue decreases by 8/3 dollars.