I cant seem to figure this one out, please help.
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.
To find the rate of increase in revenue, we need to find the derivative of the revenue function with respect to the production output x.
Given that the revenue function is R = 750 - (x^2/30), we can differentiate this function to find the rate of change as follows:
dR/dx = d(750 - (x^2/30))/dx
= d(750)/dx - d((x^2/30))/dx
= 0 - (2x/30)
= -2x/30
= -x/15
Now, we need to evaluate the rate of increase in revenue when the production is 40 kayaks per day.
Substituting x = 40 into the rate of change equation, we have:
Rate of increase in revenue = -40/15
= -8/3
Therefore, the rate of increase in revenue is -8/3 dollars per kayak per day when the production is 40 kayaks per day.
To find the rate of increase in revenue, we need to find the derivative of the revenue function with respect to time.
The revenue function is given by R = 750 - (X^2/30), where X represents the number of kayaks manufactured in one day.
To find the rate of increase in revenue, we need to differentiate this function with respect to X.
dR/dX = -X/15
Now, we know that production is increasing at a rate of 3 kayaks per day when production is 40 kayaks per day. This means that dX/dt = 3 when X = 40.
To find the rate of increase in revenue, we need to find dR/dt when X = 40.
dR/dt = (dR/dX) * (dX/dt)
We already know dX/dt = 3 and dR/dX = -X/15. Substituting these values we get:
dR/dt = (-40/15) * 3
Therefore, the rate of increase in revenue when production is at 40 kayaks per day is:
dR/dt = -8
Hence, the rate of increase in revenue is -8 dollars per day.