The area \(A\) of a circle is increasing at a constant rate of \(1.5\text{ cm}^2\text{s}^{-1}\). Find, to 3 significant figures, the rate at which the radiu...

Answer Details

To find the rate at which the radius of the circle is increasing when the area is 2 cm^2, we can use the relationship between the area and the radius of a circle.
The formula for the area of a circle is A = ?r^2, where A is the area and r is the radius.
We are given that the area of the circle is increasing at a constant rate of 1.5 cm^2/s. So, we can differentiate the area equation with respect to time to find the rate at which the area is changing.
dA/dt = 1.5 cm^2/s
Now, we can differentiate the area formula with respect to the radius to find the relationship between the rate of change of the area and the rate of change of the radius.
dA/dr = 2?r
Since we want to find the rate at which the radius is increasing, we can solve for dr/dt, which represents the rate of change of the radius.
dr/dt = (dA/dr)/(dA/dt)
dr/dt = (2?r)/(1.5 cm^2/s)
Now, we can substitute the given area value into the equation. When the area of the circle is 2 cm^2, we can find the corresponding radius value using the area formula.
2 = ?r^2
r^2 = 2/?
r = sqrt(2/?)
Now, we can substitute this value of r into the equation to find the rate at which the radius is increasing.
dr/dt = (2?(sqrt(2/?)))/(1.5 cm^2/s)
Simplifying further, we get the value of dr/dt.
dr/dt ? 0.299 cm/s
To 3 significant figures, the rate at which the radius of the circle is increasing when the area is 2 cm^2 is approximately 0.299 cm/s.
So, the correct option is 0.299 cm/s.