The radius of a blot ink is increasing at a rate of 1.55mm per seconds.find the rate at which the area is increasing after 4 seconds
To find the rate at which the area is increasing, we need to differentiate the formula for the area of a circle with respect to time.
The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.
Differentiating both sides of the equation with respect to time t, we get:
dA/dt = d/dt (πr^2)
To find the rate the area is increasing at a specific time, we substitute the given rate of change of the radius into the equation.
Given that dr/dt = 1.55 mm/s, we differentiate the equation with respect to t:
dA/dt = d/dt (πr^2)
= 2πr(dr/dt)
Substituting the given values, we have:
dA/dt = 2πr(1.55)
Now, we substitute the value of r after 4 seconds into the equation:
r = (rate of change of the radius)(time)
= (1.55 mm/s)(4 s)
= 6.2 mm
Plugging this value into our equation, we have:
dA/dt = 2π(6.2)(1.55)
≈ 60.92 mm^2/s
Therefore, the rate at which the area is increasing after 4 seconds is approximately 60.92 mm^2/s.
To find the rate at which the area is increasing, we need to differentiate the formula for the area of a circle, which is A = πr², with respect to time.
Given that the radius is increasing at a rate of 1.55 mm per second, we can express its rate of change as dr/dt = 1.55 mm/s.
Now, differentiate both sides of the area equation with respect to time (t):
d/dt(A) = d/dt(πr²)
The left side gives us the rate of change of the area, dA/dt. The right side can be found using the chain rule:
d/dt(A) = 2πr (dr/dt)
Since we are interested in the rate at which the area is increasing, substitute dr/dt = 1.55 mm/s and the given radius value at 4 seconds to solve for dA/dt at that time.
Let's assume the initial radius at t = 0 is r₀. After 4 seconds, the radius would be:
r = r₀ + (dr/dt) * t
= r₀ + (1.55 mm/s) * 4
Substitute this value of r into the rate of change equation:
dA/dt = 2πr₀ + 2π(1.55 mm/s) * 4
Calculate the numerical value to obtain the rate at which the area is increasing after 4 seconds.
a = pi r^2
da/dt = 2pi r dr/dt
If the ink blot started with radius of zero, then after 4 seconds it will have a radius of 4*1.55 = 6.2mm
da/dt = 2pi * 6.2 * 1.55 = 60.38 mm^2/sec