The profit at a lunch bar increased at a steady rate of \( 12 \% \) per year. This year the profit is \( 85,000 \mathrm{vt} \) How many years will it take the profit to exceed \( 400,000 \mathrm{vt} \) ?

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 \( P) \).

31. If \( a \) and \( b \) are such that \( 0.35 a=0.2 b \) what is \( b \) in terms of \( a \) ? A. \( 17,5 a \) B. \( 175 a \) C. \( 1,75 a \) D. \( 0,0175 a \) E. \( 0,175 a \)

a. \( 25 a^{2}+54 a b+49 b^{2} \)

23 Determine the real number \( m \) so that the equation in \( x \) would have a double root. Calculate the value of the double root. \( \mathbf{1}^{\text {o }}(m+2) x^{2}-2(m-1) x+m+2=0 \) \( \mathbf{2}^{\text {o }} 4 m x^{2}+2 x-(3 m-2)=0 \)

apendo che \( \sin \alpha=\frac{3}{5} e \cos \beta=\frac{5}{13}, \operatorname{con} \frac{\pi}{2}<\alpha<\pi e 0<\beta<\frac{\pi}{2} \), calcola: \( \cos (\alpha+\beta), \sin (\alpha-\beta), \tan 2 \alpha \)