A thin metal disk is heated altering its size, but not its shape. As the disk is heated its radius increases at a rate of 0.04 mm/sec. How fast is the disk's area changing when the radius is 225 mm

Given Information:

Radius of disk = r = 225 mm

Rate of change of radius = dr/dt = 0.04 mm/s

Required Information:

Rate of change of area = dA/dt = ?

Rate of change of area = 56.54 mm²/s = 5.7x10⁻⁵ m²/s

Explanation:

Assuming that the disk is circular shaped then the area of disk is given by

A = πr²

Where r is the radius of the disk.

Differentiating the area with respect to time t

dA/dt = πr²

dA/dt = 2πrdr/dt

Where dr/dt is the rate of change of radius

dA/dt = 2π*225*(0.04)

dA/dt = 56.54 mm²/sec

or in standard units

(0.04 mm/s)/1000 = 4.0x10⁻⁵ m/s

(225 mm)/1000 = 0.225 m

dA/dt = 2π*0.225*(4.0x10⁻⁵)

dA/dt = 5.7x10⁻⁵ m²

Therefore, the area of the disk is changing at the rate of 5.7x10⁻⁵ m²/s