If the radius of a circle increasing at the rate of 4 cm/sec, find the rate of the increase in the area when the radius is 12 cm.
a = πr^2
da/dt = 2πr dr/dt
To find the rate of increase in the area when the radius is 12 cm, we can use the derivative of the area formula in respect to time.
The formula for the area of a circle is given by:
A = πr^2
Taking the derivative with respect to time (t), we get:
dA/dt = d/dt(πr^2)
Using the power rule and the chain rule, we can differentiate the formula:
dA/dt = 2πr (dr/dt)
Substituting the given values:
dr/dt = 4 cm/sec
r = 12 cm
We can now find the rate of increase of the area by plugging in these values into the derived equation:
dA/dt = 2π(12) (4)
Simplifying the equation:
dA/dt = 96π cm²/sec
Therefore, when the radius is 12 cm, the rate of increase in the area is 96π square centimeters per second.
To find the rate of increase in the area, we need to use the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius.
We are given that the radius is increasing at a rate of 4 cm/sec, which means that dr/dt = 4 cm/sec. We want to find dA/dt, the rate of increase in the area.
To find dA/dt, we need to differentiate the equation A = πr^2 with respect to time, t.
dA/dt = d/dt(πr^2)
Using the chain rule, we can differentiate πr^2 with respect to time. Since r is a function of time:
dA/dt = d(πr^2)/dr * dr/dt
Now, let's substitute the given information into our equation. We are given that dr/dt = 4 cm/sec and we want to find dA/dt when the radius is 12 cm, so we substitute r = 12 cm:
dA/dt = d(π(12)^2)/dr * 4 cm/sec
Simplifying the equation:
dA/dt = 2πr * dr/dt
Now, substitute r = 12 cm and dr/dt = 4 cm/sec:
dA/dt = 2π(12) * 4 cm/sec
Simplifying further, we get:
dA/dt = 96π cm^2/sec
Therefore, the rate of increase in the area when the radius is 12 cm is 96π cm^2/sec.