Conically shaped pit

Clear["Global`*"]

areaSides = Pi*r*Sqrt[r^2 + h^2];
areaTop = Pi*r^2;
volume = Pi*r^2*h/3;

cost = 100*volume + 50*areaSides + 25*areaTop;

solh = Solve[volume == 50, h][[1]]

(* {h -> 150/(π r^2)} *)

The radius of the pit in meters is

solr = Solve[{D[cost /. solh, r] == 0, r > 0}, r][[1]] // N

(* {r -> 2.84461} *)

The depth of the pit in meters is

h /. solh /. solr

(* 5.90061 *)

The minimum cost in dollars is

cost /. solh /. solr

(* 8562.48 *)

Verifying,

Minimize[{cost, volume == 50, r > 0, h > 0}, {r, h}] // N

(* {8562.48, {r -> 2.84461, h -> 5.90061}} *)