At time $t=0,$ a boiled potato is taken from a pot on a stove and left to cool in a kitchen. The internal temperature of the potato is $91$ degrees Celsius $(C)$ at time $t=0,$ and the internal temperature of the potato is greater than $27C$ for all times $t>0.$ The internal temperature of the potato at time $t$ minutes can be modelled by the function $H$ that satisfies the differential equation $\frac{dH}{(d)t}=-\frac{1}{4}(H-27)$ where $H(t)$ is measured in degrees Celsius and $H(0)=91.$

$($i$)$ Write an equation for the tangent to the graph of $H$ at $t=0.$ Use this equation to approximate the internal temperature of the potato at time $t=3.$

$($ii$)$ Use $\frac{d^{2}H}{(d)t^2}$ to determine whether your answer in $($i$)$ is an underestimate or overestimate of the internal temperature of the potato at time $t=3.$

$($iii$)$ For an alternative model for the internal temperature of the potato at time $t$ minutes is the function $G$ that satisfies the differential equation $\frac{dG}{(d)t}=-(G-27{)}^{\frac{2}{3}}$ where $G(t)$ is measured in degrees Celsius and $G(0)=91.$ Find an expression for $G(t).$ Based on this model find the internal temperature of the potato at time $t=3\mathrm{.}9$