A sphere of radius r, density,`rho` and specific heat s is heated to temperature `theta` and then cooled in an enclosure at temperature `theta_(0)` . How , the rate of fall of temperature is related with r, `rho` and s?

Explanation

`(dQ)/(dt) = KA(d theta)/(dx) or d/(dt) (ms theta) = KA (d theta)/(dx)`
or `ms (d theta)/(dt) = KA (d theta)/(dx) or (d theta)/(dt) = (KA)/(ms) (d theta)/(dx)`
or, `(d theta)/(dt) prop (A)/(ms)`
`( :' K(dtheta)/(dx)` is constant as per question)
Since `A prop r^(2)` and `m prop r^(3) rho `, so
We can write, `(d theta)/(dt) prop (r^2)/(r^(3)rho s) or (d theta)/(dt) prop 1/(r rho s)`.