Corrigés — Étude de signes dans des modèles réels#
Trajectoire d’une balle#
On résout :
\(12t - 4.9t^2 > 5\)
\(-4.9t^2 + 12t - 5 > 0\)
Discriminant : \(\Delta = 46\).
Racines :
\(t_1 \approx 0.51,\qquad t_2 \approx 2.00\)
Tableau de signes#
| \(t\) | \(-\infty\) | \(0.51\) | \(2\) | \(+\infty\) | |
| \(t - 0.51\) | \(-\) | \(0\) | \(+\) | \(+\) | |
| \(t - 2\) | \(-\) | \(-\) | \(0\) | \(+\) | |
| \(-4.9\) | \(-\) | \(-\) | \(-\) | ||
| \(\text{Produit}\) | \(-\) | \(0\) | \(+\) | \(0\) | \(-\) |
Conclusion#
\([0.51 < t < 2.00]\)
Rentabilité d’un artisan#
On résout :
\(B(x)= -0.2x^2 + 14x - 100 > 0\)
Discriminant : \(\Delta = 2900\)
Racines :
\(x_1 \approx 9.13,\qquad x_2 \approx 54.87\)
Tableau de signes#
| \(x\) | \(-\infty\) | \(9.13\) | \(54.87\) | \(+\infty\) | |
| \(x-9.13\) | \(-\) | \(0\) | \(+\) | \(+\) | |
| \(x-54.87\) | \(-\) | \(-\) | \(0\) | \(+\) | |
| \(-0.2\) | \(-\) | \(-\) | \(-\) | ||
| \(\text{Produit}\) | \(-\) | \(0\) | \(+\) | \(0\) | \(-\) |
Conclusion#
\([9 \le x \le 55]\)
Débit d’un bassin pluvial#
Numérateur nul : \(h=20\)
Tableau de signes#
| \(h\) | \(0\) | \(20\) | \(+\infty\) |
| \(h-20\) | \(-\) | \(0\) | \(+\) |
| \(h+5\) | \(+\) | \(+\) | |
| \(\text{Quotient}\) | \(-\) | \(\texttt{X}\) | \(+\) |
Conclusion#
\([0 \le h < 20]\)