1126-1128

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[aleksandar422]

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\documentclass{article} \usepackage[utf8]{inputenc} \usepackage[serbian]{babel} \usepackage{tikz} \usepackage{amsmath, amssymb, amsthm}

\newtheorem{zad}{Zadatak}

\renewcommand{\figurename}{Slika} \newcommand{\Placeholder}[2][(10, 10)] { \begin{tikzpicture} \draw[help lines] (0, 0) grid #1; \pgfgetlastxy{\x}{\y} \node[rotate=-25, align=center] at (\x / 2, \y / 2) {{\Huge Placeholder} \ #2}; \end{tikzpicture} }

\begin{document} \begin{zad} \quad$2)\quad (ab) = \frac{4}{5}m, \quad (cd) = 1\frac{5}{8}, \quad (ad) = 0.6, \quad P(abcd) = \textrm{?}$ \end{zad} \begin{align} &(ab) = \frac{4}{5}\, m, \quad (cd) = 1\frac{5}{8} = \frac{13}{8}\, m, \quad (ad) = 0.6\, m = \frac{3}{5}m \ &P = \frac{(ab) + (cd)}{2} \cdot (ad) = \frac{\frac{4}{5} + \frac{13}{8}}{2} \cdot \frac{3}{5} \, m^2 = \frac{291}{400} \, m^2 = 0.7275 \, m^2 \ &\textrm{ ili } (ab) = \frac{4}{5}\, m = 0.8\, m; \quad (cd) = \frac{13}{8}\, m = 1.625\, m; \quad (ad) = 0.6\, m \ &P = \frac{(ab) + (cd)}{2} = \frac{0.8 + 1.625}{2} \cdot 0.6 = 2.425 \cdot 0.3 = 0.7275 \, m^2 \end{align}

\begin{zad}
    \quad$1)\quad (aa') = 3.5\, cm, \quad (mn) = 2.4\, cm$
    \begin{align*}
        &P(aa'b'b) = [(aa')\cdot(mn)]kj = (3.5 \cdot 2.4)\, cm^2 = (3.5 \cdot 2.4)\, cm^2 = 8.4\, cm^2
    \end{align*}
    \indent \phantom{Zadatak. } \quad$2)\quad (a'b') = 10.4\, cm \quad (ef) = 5.2\, cm$
    \begin{align*}
        &P(aa'b'b) = [(a'b') \cdot (ef)]kj = (10.4 \cdot 5.2)\, cm^2 = 54.08\, cm^2
    \end{align*}
\end{zad}

\begin{zad}
    \quad$n = 2, \quad \alpha = 36 ^\circ, \quad a = 2r\sin\alpha, \quad h = r\cos \alpha$
    \begin{align*}
        &P_5 = na \cdot\frac{h}{2} = \frac{5}{2}ah \\
        &a = 2r\sin 36^\circ = 2r \cdot 0.588 = 1.176r, \quad h = r\cos 36^\circ = r \cdot 0.809 = 0.809r, \\
        &a = 1.176r \quad \textrm{ i }  \quad h = 0.809r \\
        &P_5 = \frac{5}{2}1.176r \cdot 0.809r \approx 2.38 r^2
    \end{align*}
    \indent \phantom{Zadatak. } \quad $n = 8, \quad \alpha = 22^\circ 30'; \quad a = 2r\sin20^\circ30' = 2r\cdot0.382 = o.764r \\
    \quad h = r\cos22^\circ30' = r\cdot0.923 = 0.923r$
    \begin{align*}
        P_8 = 8a\cdot\frac{h}{2} = 4ah = 4\cdot0.764r\cdot0.932r \approx 2.83r^2
    \end{align*}
    \indent \phantom{Zadatak. } \quad $n = 12, \quad \alpha = 15^\circ, \quad a = 2r\sin15^\circ = 2r\cdot0.254 = 0.518r \\
    \quad h = r\cos15^\circ = r\cdot0.966 = 0.966r$
    \begin{align*}
        P = 12a\frac{h}{2} = 6ah = 6\cdot0.518r\cdot0.966r \approx 3r^2
    \end{align*}
\end{zad}

\begin{zad}
    \quad $n = 5, \quad \alpha = 36^\circ, \quad a = 2r\tg x, \quad h = r$
    \begin{align*}
        a &= 2r\tg 36^\circ = 2r\cdot0.726 = 1.452r, \quad h = r \\
        P_5 &= 5a\frac{h}{2} = \frac{5}{2}ah = \frac{5}{2}1.425r\cdot r = 3.64r^2
    \end{align*}
    \indent \phantom{Zadatak. } \quad$n = 8, \quad \alpha = 22^\circ30';\quad a = 2r\arctg x, \quad h = r$
    \begin{align*}
        a &= 2r\tg 22^\circ 30' = 2r\cdot 0.414 = 0.828r, \quad h = r \\
        P_8 &= 8a\cdot \frac{h}{2} = 4ah = 4\cdot 0.828r\cdot r \approx 3.31r^2
    \end{align*}
    \indent \phantom{Zadatak. } \quad $n = 12, \quad \alpha = 15^\circ, \quad a = 2r\tg \alpha, \quad h = r$
    \begin{align*}
        a &= 2r\tg 15^\circ = 2r\cdot 0.268 = 0536r, \quad h = r \\
        P_{12} &= 12a\frac{h}{2} = 6ah = 6\cdot 0.536r\cdot r = 3.216 r^2 \approx 3.22r^2
    \end{align*}
\end{zad}

\begin{zad}
    \quad $q = 175\pi\, \, cm^2, \quad q = \textrm{omota\v c}, \quad r = 7\, cm$
    \begin{align*}
        q &= \pi r s \\
        \pi r s &= 175\pi \\
        \pi s &= 175 \\
        7s &= 175 \Rightarrow s = 25
    \end{align*}    
\end{zad}
\begin{figure}[h]
    \center
    \Placeholder[(3, 3)]{Ovde stoji opis ovog crteza}
    \caption{}
\end{figure}
\begin{align*}
    h^2 &= s^2 - r^2 = 25^2 - 7^2 = (25 - 7)(25 + 7) = 18 \cdot 32 = 576 \\
    h &= \sqrt{576} = 24 \quad \Longrightarrow \quad h = 24\, cm \\
    P &= \pi r(r + s) = \pi \cdot 7 (7 + 25) = \pi \cdot 7 \cdot 32 = 224 \pi \\
    &= \frac{1}{3}r^2 \pi h = \frac{1}{3} 7^2 \cdot \pi \cdot 24 = 49 \cdot 8\pi = 392 \pi \quad \Longrightarrow \quad V = 392 \pi \, cm^3 
\end{align*}

\begin{zad}
    \quad $P_p = 48 \pi\, cm^2$
\end{zad}
\begin{figure}[h]
    \center
    \Placeholder[(3, 3)]{Ovde stoji opis ovog crteza}
    \caption{}
\end{figure}

$\bigtriangleup Oab$ je jednakostrani\v can
\begin{align*}
    P_p &= 48\pi \\
    {r_1}^2 \pi &= 48\pi \\
    {r_1}^2 &= 48 \quad \longrightarrow \quad r_1 = \sqrt{48} = 4\sqrt{3} \\  
\end{align*}

iz $\bigtriangleup OAO_1$ je ${r_1}^2 + \frac{r}{2}^2 = r^2$
\begin{align*}
    {r_1}^2 &= r^2 - \frac{r}{2}^2 = r^2 \\
    r_1 &= \sqrt{\frac{3}{4}r^2} = \frac{r\sqrt{3}}{2} \\
    \frac{r\sqrt{3}}{2} &= 4\sqrt{3} \quad \Longrightarrow \quad \frac{r}{2} = 4 \quad \Longrightarrow \quad r = 8\, cm \\
    P &= 4r^2 \pi = 4 \cdot 8^2 \pi = 4 \cdot 64 \pi = 256\pi \\
    P &= 256\pi \, cm^2, \\
    V &= \frac{4}{3}r^3 \pi = \frac{4}{3} 8^3 \pi = \frac{4}{3} 512 \pi = \frac{2048}{3} \pi \\
    V &= \frac{2048}{3} \pi \, cm^3.
\end{align*}

\end{document}

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[aleksandar422]

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