1137 - Expanding Rods.cpp

/*8<{==========~ BEGIN TEMPLATE ~============>8*/
#include <bits/stdc++.h>
using namespace std;
#ifdef LOCAL
#include "debug.cpp"
#else
#define dbg(...)
#endif
#define endl '\n'
#define fastio                  \
  ios_base::sync_with_stdio(0); \
  cin.tie(0);
#define all(j) j.begin(), j.end()
#define rall(j) j.rbegin(), j.rend()
#define len(j) (int)j.size()
#define rep(i, a, b)                           \
  for (common_type_t<decltype(a), decltype(b)> \
           i = (a);                            \
       i < (b); i++)
#define rrep(i, a, b)                          \
  for (common_type_t<decltype(a), decltype(b)> \
           i = (a);                            \
       i > (b); i--)
#define trav(xi, xs) for (auto &xi : xs)
#define rtrav(xi, xs) \
  for (auto &xi : ranges::views::reverse(xs))
using ll = long long;
#define inte ll
#define pb push_back
#define pf push_front
#define ppb pop_back
#define ppf pop_front
#define eb emplace_back
#define lb lower_bound
#define ub upper_bound
#define fi first
#define se second
#define emp emplace
#define ins insert
#define divc(a, b) ((a) + (b)-1ll) / (b)
using str = string;
using ull = unsigned long long;
using ld = long double;
using vll = vector<ll>;
using pll = pair<ll, ll>;
using vll2d = vector<vll>;
using vi = vector<int>;
using vi2d = vector<vi>;
using pii = pair<int, int>;
using vpii = vector<pii>;
using vc = vector<char>;
using vs = vector<str>;
template <typename T, typename T2>
using umap = unordered_map<T, T2>;
template <typename T>
using pqmn =
    priority_queue<T, vector<T>, greater<T>>;
template <typename T>
using pqmx = priority_queue<T, vector<T>>;
template <typename T, typename U>
inline bool chmax(T &a, U const &b) {
  return (a < b ? a = b, 1 : 0);
}
template <typename T, typename U>
inline bool chmin(T &a, U const &b) {
  return (a > b ? a = b, 1 : 0);
}
/*8<============~ END TEMPLATE ~============}>8*/
void run();

int32_t main() {
#ifndef LOCAL
  fastio;
#endif
  cout<<fixed<<setprecision(10);

  int T = 1;

  cin >> T;

  rep(t, 0, T) {
    cout<<"Case "<<t+1<<": ";
    dbg(t);
    run();
  }
}
const ld EPS=1e-9;

ld chordLenght(ld r, ld theta){
  return 2.0 * r * sin(theta / 2.0);
}
template<typename T>
bool equals(T a, T b) {
  if (std::is_floating_point<T>::value)
    return fabs(a - b) < EPS;
  else
    return a == b;
}
void run() {
  ld chord,n,c;cin>>chord>>n>>c;
  ld arc= (1+n*c)*chord;
  ld low=1e-9;
  ld high=1e18;
  ld defr;
  for(;;){
    auto r=(low+high)/2;
    ld theta=arc/r;
    ld curchord=chordLenght(r,theta);
    if(equals(curchord,chord)){
      defr=r;
      break;
    }
    if(curchord<chord)
      low=r;
    else high=r;
  }
  ld theta=arc/defr;
  ld h=defr*(1-cos(theta/2));
  cout<<h<<endl;
}

/*8<
  AC, geometry
>8*/