A circular ink blot grows at the rate of 2 cm per sec. Find the rate at which the radius is increasing after 28/11 sec.
Well, that's quite an interesting question! We can find the rate at which the radius is increasing using a little calculus.
Let's start by noting that the rate at which the circular ink blot is growing is given as 2 cm per second. This rate refers to the rate at which the area of the ink blot is increasing.
Now, we know that the area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.
To find the rate at which the radius (r) is increasing, we need to differentiate both sides of the equation with respect to time (t).
dA/dt = d/dt (πr^2)
Now, let's use the chain rule to differentiate the right side of the equation:
dA/dt = 2πr(dr/dt)
We're given that dA/dt (the rate at which the area is increasing) is 2 cm^2/sec, and we want to find dr/dt (the rate at which the radius is increasing) after 28/11 seconds.
So, let's plug in the given information into our equation:
2 = 2πr(dr/dt)
And since we're interested in finding dr/dt after 28/11 seconds, we substitute r = 28/11 into the equation:
2 = 2π(28/11)(dr/dt)
Now we can solve for dr/dt:
dr/dt = 2/(2π(28/11))
Simplifying further:
dr/dt = 11/(2π(28/11))
dr/dt ≈ 0.196 cm/sec
So, after all that calculation, we find that the rate at which the radius is increasing after 28/11 seconds is approximately 0.196 cm/sec. Voila!
To find the rate at which the radius is increasing, we need to differentiate the equation that relates the radius and time.
Let's assume the radius of the circular ink blot at any given time t is r and the rate at which it grows is dr/dt.
We know that the circumference of a circle is given by the formula C = 2πr. Given that the ink blot is growing at a rate of 2 cm per sec, we can say that dC/dt = 2 cm/sec.
Now, we can differentiate the circumference equation with respect to time (t) to find dr/dt, the rate at which the radius is increasing:
dC/dt = d(2πr)/dt
2 cm/sec = 2π (dr/dt)
Simplifying the equation, we get:
dr/dt = 1/π cm/sec
This tells us that the rate at which the radius is increasing is 1/π cm/sec.
To find the rate at which the radius is increasing after 28/11 sec, substitute t = 28/11 sec into the equation:
dr/dt = 1/π cm/sec
dr/dt = 1/π cm/sec (when t = 28/11 sec)
After solving the equation, you will find the rate at which the radius is increasing after 28/11 sec.
πr^2=2×28/11=56/11,r=14/11
Now a=πr^2
And derivative w.r.t t,we get
da/dt=2πr dr/dt
2=2π×14/11×dr/dt
dr/dt=11/14π=1/4=.25
πr^2=2×28/11
r=14/11
let,a=Ï€r^2
da/dt=2Ï€r dr/dt
2 =2π×14/11×dr/dt
dr/dt=11/14Ï€ =1/4
=.25
I will assume you meant to say that the area of the ink blot is growing at the rate of 2 cm^2 /s
A = πr^2
dA/dt = 2πr dr/dt
2 = 2π (28/11) dr/dt
dr/dt = 11/(28π) cm/sec
= appr .125 cm/s