When designing a roadway for vehicle safety the maximum permissible curvature at any point is $ k_{max} = 0.002. $ If we consider that a curved section of this road is described by the function $ y = a x^2, $ with $ x \in (-10,10), $ I have to find the value of the factor $ a $ so that the maximum curvature is within the safe limit. I know that the curvature is $ \sqrt {x^{''2}+y{''2}} =1$ and $k = \dfrac{|y''|}{(1 + y'^2)^{3/2}} $ but I can't apply this equation to my problem.
Secondly, we consider a metal sphere with a radius of $ a $ and a temperature of $ T_0, $containing a static fluid. The sphere heats the fluid environment, with the result that the temperature distribution of the fluid surrounding the sphere is given by the relation $$ T = T_0 [\frac {x ^ 2 + y ^ 2 + z ^ 2} {a ^ 2}] ^ {\frac {1} {3}}, $$ where $ x ^ 2 + y ^ 2 + z ^ 2 \ge a ^ 2. $ I have to calculate the derivative of the temperature in the direction perpendicular to the surface of the sphere $ \frac {dT} {dn} $ and find the rate of heat transfee. I don't know how to start
First part $$y= a x^2 =\dfrac{x^2}{2R}==\dfrac{k_{max} x^2}{2}$$
Calculation of curvature from formula
$$ \dfrac{1}{R}=\dfrac{y''}{(1+y^{'2})^{3/2}}$$
$ y=a x^2,y'=2a x, y''=2a; $ if $x=0,y=0,y'=0, y''=2a/(1+0)^{\frac32}=2a $
Second part
$$\dfrac{T}{T_0}=r^\frac32 $$
$$\dfrac{dT}{dr}=\frac{3T_0}{2} \;r^\frac12 $$