Calcola il perimetro e l'area del triangolo \( A B C \) della figura a lato avente il seg mento \( A H \) che misura 25 cm , la base \( A B \) che misura 40 cm e l'angolo \( \widehat{A} \) am pio \( 120^{\circ} \).

Scienze Un recipiente cilindrico contiene un miscuglio formato da limatura di ferro e polvere di zolfo. La polvere di zolfo ha una massa di 150 g , mentre la massa della limatura di ferro è \( \frac{1}{3} \) della massa totale. Calcola la massa dellali. matura di ferro e quella del miscuglio.

tercice 5. Soit \( n \in \mathbb{N}^{*} \). On définit la relation \( \mathcal{R} \) sur \( \mathbb{C} \) par: \( z \mathcal{R} z^{\prime} \Leftrightarrow z^{n}=z^{\prime n} \) Montrer que \( \mathcal{R} \) est une relation d'équivalence. Soit \( \mathrm{a} \in \mathbb{C}^{*} \). Déterminer la classe d'équivalence de a note \( \mathrm{cl}(\mathrm{a}) \). En déduire les classes \( \mathrm{cl}(1+\mathrm{i}) \) et \( \mathrm{cl}(-1+i \sqrt{3}) \) pour \( \mathrm{n}=3 \) et \( \mathrm{n}=4 \).

Volume of a Tank A civil engineering firm is tasked with designing a new water storage tank for a municipal water supply system. The engineers want the tank to have an efficient shape that maximizes storage capacity while minimizing the amount of construction material used. The tank's cross-section will be shaped by revolving the regions bounded by the lines and curves, \( x=\sqrt{2 \sin ^{2} y} \) \( 0 \leq y \leq \frac{\pi}{2}, x=0 \) about the \( y \)-axis. The goal is to determine the volume of the tank that will be created by this revolution around the \( y \)-axis. 4.1 Source: Mafemba, J. (2024) a. Calculate the volume of the solid generated by revolving the regions bounded by the lines and curves mentioned in the scenario above

24 Consider the equation of the second degree in \( x: x^{2}-(m+3) x+m=0 \), where \( m \) is a real parameter. Show that there is between the roots, when they exist, an independent relation from \( m \). (Eliminate \( m \) between \( S \) and \( P) \).

1. Summarize the events of the battle of Soissons