4. When the area in square units of an expanding circle is increasing twice as fast as its radius in linear units, the radius is

Question

4. When the area in square units of an expanding circle is increasing twice as fast as its radius in linear units, the radius is

A. 1/4pi B.1/4 C.1/pi D. 1 E.pi

Answer 1

Area = πr²

dA/dr=2πr
dA/dt=dA/dr*dr/dt=2πrdr/dt
So when dA/dt=2dr/dt
2πrdr/dt=2dr/dt
r=1/π

Answer 2

Well, it seems like this expanding circle has a sense of humor! The area increasing twice as fast as the radius? That's quite an overachiever. Now, let's figure out its sense of proportion.

Let's call the radius of the expanding circle "r" (since it loves to keep things simple and short). We know that the formula for the area of a circle is A = πr².

According to the question, the area is increasing twice as fast as the radius. So, we can say that dA/dt = 2dr/dt.

Differentiating both sides of the area formula with respect to time, we get dA/dt = 2πr(dr/dt).

Now we can substitute dA/dt = 2dr/dt (given in the question) into the equation and simplify:

2dr/dt = 2πr(dr/dt)
dr/dt = πr(dr/dt)

Now, we can divide both sides of the equation by (dr/dt):

1 = πr

Since we're solving for the radius (r), the answer is D. 1. The radius of this expanding circle really likes to keep things simple and straightforward.

Answer 3

Let's denote the radius of the expanding circle as "r" and the area of the circle as "A".

The formula for the area of a circle is A = πr^2.

According to the given information, the area of the circle is increasing twice as fast as its radius. This can be expressed mathematically as:

dA/dr = 2

Now, we can differentiate the area formula with respect to the radius to find the rate of change of the area with respect to the radius:

dA/dr = 2πr

Setting this equal to 2, we have:

2πr = 2

Dividing both sides by 2π:

r = 1/π

Therefore, the radius of the expanding circle is 1/π.

So, the answer is C. 1/π.

Answer 4

To solve this problem, we need to use the formula for the area of a circle. The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.

We know that the area of the circle is increasing twice as fast as its radius. In other words, the rate of change of the area (dA/dt) is twice the rate of change of the radius (dr/dt).

Using calculus, we can differentiate the equation for the area of a circle with respect to time (t):

dA/dt = 2πr(dr/dt)

Now we can plug in the given information that the rate of change of the area (dA/dt) is twice the rate of change of the radius (dr/dt):

2(dr/dt) = 2πr(dr/dt)

Canceling the 2's gives us:

dr/dt = πr(dr/dt)

Now we can cancel out the (dr/dt) terms:

1 = πr

To solve for the radius (r), divide both sides of the equation by π:

r = 1/π

Therefore, the correct answer is C. 1/π.