Question

The area of a sector in a circle is given by the formula: A=12r2θ , where r is the radius and θ is the central angle measured in radians. Find the rate of change of A with respect to θ if r remains constant. What is the rate when r=2?

Answer 1

Answer:

Step-by-step explanation:

$A=\frac{1}{2} r^{2} \theta\\\frac{dA}{d\theta} =r \\\\when r=2\\rate of change at r=2 is  2$

Answer 2

We want to find the rate of change of a given function of two variables when we fix one of the two, and then we want to evaluate it at r = 2.

We will get that the rate of change is:

$\frac{dA(r, \theta)}{d\theta} = 12*r^2$

And when r = 2, the rate is 48.

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We define the rate of change of a function with respect to some variable as the differentiation of the function with respect to that variable.

Here we have the function:

$A(r, \theta) = 12*r^2*\theta$

We need to differentiate it with respect to θ, we will get:

$\frac{dA(r, \theta)}{d\theta} = 1*12*r^2*\theta^0 = 12*r^2$

Where I used the general rule to derive functions with exponents:

$f(x) = x^n\\\\\frac{df(x)}{dx} = n*x^{n -1}$

Now that we know the rate of change, we want to evaluate it in r = 2, we will get:

$\frac{dA(2, \theta)}{d\theta} = 12*2^2 = 48$

Notice that this does not depend on the value of θ.

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