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.
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To find the rate of increase in revenue, we need to find the derivative of the revenue function with respect to time. In this case, time is represented by the production output, denoted by x.
The revenue function is given by:
R = 750 - (x^2 / 30)
To find the derivative of this function, we can use the power rule. The power rule states that for a function of the form f(x) = kx^n, the derivative is given by f'(x) = nkx^(n-1).
Differentiating the revenue function with respect to x:
dR/dx = (d/dx)(750 - (x^2 / 30))
The derivative of 750 with respect to x is zero since it is a constant. For the second term, we apply the power rule:
dR/dx = 0 - (2x / 30)
Simplifying the expression:
dR/dx = -2x / 30
Now, we need to find the rate of increase in revenue when production is 40 kayaks per day. This means we need to evaluate the derivative at x = 40:
dR/dx = -2(40) / 30
Simplifying further:
dR/dx = -80 / 30
Dividing -80 by 30:
dR/dx = -8/3
Therefore, the rate of increase in revenue when production is 40 kayaks per day is -8/3 dollars per kayak per day.