To determine the ratio of the increase in the surface areas of the cube and the sphere, we will differentiate their respective surface areas with respect to time and then compare the rates of change.
First, let's denote the side length of the cube as
x and the radius of the sphere as
r. It is given that the side length of the cube is equal to the diameter of the sphere, so we have:
x=2r We know that the surface area of a cube is given by:
A‌cube ‌=6x2Differentiating both sides with respect to time
t, we get:
‌=6⋅2x⋅‌=12x‌The surface area of a sphere is given by:
A‌sphere ‌=4πr2 Differentiating both sides with respect to time
t, we get:
‌=4π⋅2r⋅‌=8πr‌Given that the side of the cube and the radius of the sphere increase at the same rate, we have:
‌=‌ Let's denote this common rate of increase as
k :
‌=‌=kNow, substituting
x=2r into the expression for the rate of increase of the cube's surface area, we get:
‌=12xk=12⋅2rk=24rk And for the sphere:
‌=8πrkFinally, we take the ratio of these two rates of increase:
‌=‌=‌=‌So, the ratio of the increase in their surface areas is:
3:Ï€Therefore, the correct answer is Option A.