12. A cylindrical metal pipe has external diameter of 6 cm and internal diameter of 4 cm . Calculate the volume of metal in a pipe of length 1 m . If \( 1 \mathrm{~cm}^{3} \) of the metal has a mass of 8 g , find the mass of the pipe. 13. For two cylinders A and B , the ratio of lengths is \( 3: 1 \) and the ratio of diameters is \( 1: 2 \). Calculate the ratio of their volumes. 14. A machine makes boxes which are either perfect cylinders of diameter and length 4 cm , or perfect cubes of side 5 cm . Which boxes have the greater volume, and by how much? (Take \( \pi=3 \) ) 15. Natalia decided to build a garage and began by calculating the number of bricks required. The garage was to be 6 m by 4 m and 2.5 m in height. Each brick measures 22 cm by 10 cm by 7 cm . Natalia estimated that she would need about 40000 bricks. Is this a reasonable estimate?

84 On considère la suite \( \left(u_{n}\right) \) définie par \( u_{0}=0 \) et, pour tout entier naturel \( n, u_{n+1}=3 u_{n}-2 n+3 \). 1. Calculer \( u_{1} \) et \( u_{2} \). 2. Démontrer par récurrence que, pour tout entier naturel \( n, u_{n} \geqslant n \). 3. En déduire que la suite \( \left(u_{n}\right) \) est croissante. 4. Soit la suite \( \left(v_{n}\right) \) définie, pour tout entier naturel \( n \), par \( v_{n}=u_{n}-n+1 \). a. Démontrer que la suite \( \left(v_{n}\right) \) est une suite géomé- trique de raison 3 . b. En déduire que, pour tout entier naturel \( n \) : \( \quad u_{n}=3^{n}+n-1 \). c. Calculer \( \sum_{k=0}^{100} u_{k}=u_{0}+u_{1}+\ldots+u_{100} \). On rappelle que, pour tout entier naturel \( n \) : \( 1+2+\ldots+n=\frac{n(n+1)}{2} \) et, pour \( q \neq 1 \) : d. ALco Écrire un algorithme en langage naturel qui calcule la somme ci-dessus.

If \( \sin A+\cos A=\frac{1}{4} \) find \( \sin A \cos A \)

Stabilisci per quale valore di \( a \) le rette di equazio- ni \( x-y+a=0 \) e \( x-3 y=0 \) si intersecano in un punto di ascissa 3

Mid Aerm Assignment. if \( 2, x, y,-230 \ldots \) a geogrametric progr Find \( x \) and \( y \).

Составить уравнение параболы с вершшной в точке \( \mathrm{Ol}(21 ;-2) \) и с фокусом в точке \( \mathrm{F}(2 ;-2) \)