The radius of a circle in increasing at the rate of 0.5cm/s. Find the rate at which the circumference increases if the radius in 5cm.
C = 2πr
dC/dt = 2π dr/dt
= 2π(.5) cm/s
= π cm/s
To find the rate at which the circumference increases, we need to differentiate the formula for the circumference of a circle with respect to time.
The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.
Differentiating both sides of the equation with respect to time gives us:
dC/dt = d/dt (2πr)
Recall that the derivative of a constant multiplied by a function is equal to the product of the constant and the derivative of the function. Therefore, the derivative of 2πr with respect to time is:
dC/dt = 2π(d/dt (r))
Given that the radius is increasing at a rate of 0.5 cm/s, we can substitute this value:
dC/dt = 2π(d/dt (5 cm))
Taking the derivative of the constant 5 cm with respect to time, we have:
dC/dt = 2π(0 cm/s)
Since the rate of change of a constant is always zero, the last term becomes zero:
dC/dt = 0
Thus, the rate at which the circumference increases is 0 cm/s.
To find the rate at which the circumference increases, we can use the formula for the circumference of a circle:
Circumference = 2πr
where r is the radius.
Differentiating both sides of the equation with respect to time will give us the rate of change of the circumference:
dCircumference/dt = d(2πr)/dt
Since the radius is increasing at a constant rate of 0.5 cm/s, we can substitute this value in for dr/dt:
dCircumference/dt = d(2π(5))/dt
Simplifying the equation, the derivative of a constant value such as 2π is zero:
dCircumference/dt = 0
Therefore, the rate at which the circumference increases is 0 cm/s.