A generator produces electrical power, P, in watts, according to the function:
P(R)= 120/ (0.4 + R)^2
where R is the resistance, in ohms. Determine the intervals on which the power is increasing.
No answer, not solvable
To determine the intervals on which the power is increasing, we need to find the derivative of the power function with respect to the resistance (R) and then analyze the sign of the derivative.
The power function is given as: P(R) = 120/ (0.4 + R)^2
To find the derivative of P(R) with respect to R, we can use the quotient rule. Let's denote P'(R) as the derivative of P(R):
P'(R) = (d/dR)[120/ (0.4 + R)^2]
Using the quotient rule:
P'(R) = [ (0.4 + R)^2 * 0 - 120 * 2 * (0.4 + R) ] / (0.4 + R)^4
Simplifying:
P'(R) = (-240 * (0.4 + R)) / (0.4 + R)^4
We can further simplify by canceling out common factors:
P'(R) = -240 / (0.4 + R)^3
Now, to determine the intervals on which the power is increasing, we need to analyze the sign of the derivative, P'(R).
Since the numerator of the derivative is a negative constant (-240), the sign of the derivative will be determined by the denominator, (0.4 + R)^3.
For the denominator (0.4 + R)^3, it is positive for all values of R, because raising a positive number to any power will result in a positive value.
Therefore, the sign of P'(R) will be negative for all values of R. This means that the power function P(R) is decreasing for all values of R.
As a result, there are no intervals in which the power is increasing based on the given power function.