Let A be the area of a circle with radius r. If dr/dt=4, find dA/dt when r=1.

Question

Let A be the area of a circle with radius r. If dr/dt=4, find dA/dt when r=1.

Answer 1

Let's start with what we know about the radius:

dr/dt = 4

Integrating both sides, we get:

r = 4t

I'm going to assume that the initial length of the radius at time 0 is 0, since it's not specified in the problem.

From this, we can figure out the time t when the radius is 1:

1 = 4t

t = 1/4

Now, let's look at area. The formula for the area of a circle is:

A = π r²

We can rewrite this as a function of t since we know r=4t from above:

A = π (4t)² = 16 π t²

Now let's take the derivative to get an equation of the rate of change of the area with respect to time:

A'(t) = 32 π t

Plugging in t = 1/4 (which is when the radius is 1), we get:

A'(1/4) = 32 π (1/4) = 8 π

So the rate of change is 8 π per unit of time (whatever that is, since it isn't specified in the problem).

Answer 2

Hey Jaswinder,

This question is testing your understanding of related rates!

So, we know that the area of a circle is given by

A = π r²

To determine how A changes with time, we take the rate of change (derivative)

dA/dt = d/dt (π r²)

We must now differentiate the right hand side. Being that π is constant, you pull it out of the d/dt operation.

dA/dt = π*d/dt (r²)

Now, since r is a function of time, we must apply the chain rule.

d/dt (r²) = 2*r* dr/dt

Now, by substituting the expressions we have found above, we can write

dA/dt = d/dt (π r²) ------> dA/dt = 2*π*r* dr/dt

Since it is given that dr/dt = 4, and r = 1,

dA/dt = 2*π*1*4

dA/dt = 8π

Hope that helps!

-Marco