Εμβαδόν γραμμοσκιασμένης περιοχής

Tolaso J Kos · #1

Να υπολογιστεί το εμβαδόν της παρακάτω γραμμοσκιασμένης περιοχής.

$\displaystyle{\begin{tikzpicture} %apply color to the canvas \draw[dashed, fill=magenta] (-4, 0) -- (4, 0) -- (4, 8) -- (-4, 8) -- cycle; \draw[<->, line width=1.2pt] (4.3, 2) -- (4.3, 6); \draw (4.3, 4) node[right]{5}; %apply white color to the semicircles \draw[samples=1000, fill=white] (2, 0) arc(0:180:2cm); \draw[samples=1000, fill=white] (4, 6) arc(90:270:2cm); \draw[samples=1000, fill=white] (-4, 6) arc(90:-90:2cm); \draw[samples=1000, fill=white] (-2, 8) arc(-180:0:2cm); %fill the arcs with white color \fill[white] (-4, 2) arc(90:0:2cm) -- (-4, 2) -- (-4, 0) -- (-2, 0); \fill[white] (4, 2) arc(90:180:2cm) -- (4, 2) -- (4, 0) -- (2, 0); \fill[white] (2, 8) -- (4, 6) arc(-90:-180:2cm) -- (4, 6) -- (4, 8); \fill[white] (-4, 6) -- (-4, 6) arc(-90:0:2cm) -- (-2, 8) -- (-4, 8); %fill the interior arcs with white color \fill[white, domain=-2:0] (0, 6) arc(0:-90:2cm) -- plot(\x, {sqrt(4-(\x)^2)+4} ); \fill[white, domain=0:2] (2, 4) arc(270:180:2cm) -- plot(\x, {sqrt(4-(\x)^2)+4} ); \fill[white, domain=0:2] (2, 4) arc(90:180:2cm) -- plot(\x, {-sqrt(4-(\x)^2)+4} ); \fill[white, domain=-2:0] (-2, 4) arc(90:0:2cm) -- plot(\x, {-sqrt(4-(\x)^2)+4} ); %draw the central circle \draw (0,4) circle(2cm); %draw the arcs \draw[samples=1000] (-4, 2) arc(90:0:2cm); \draw[samples=1000] (4, 2) arc(90:180:2cm); \draw[samples=1000] (4, 6) arc(-90:-180:2cm); \draw[samples=1000] (-4, 6) arc(-90:0:2cm); %draw the arcs in the interior of the circle \draw[samples=1000] (0, 6) arc(0:-90:2cm); \draw[samples=1000] (2, 4) arc(90:180:2cm); \draw[samples=1000] (-2, 4) arc(90:0:2cm); \draw[samples=1000] (2, 4) arc(270:180:2cm); \end{tikzpicture}}$

#2

Tolaso J Kos έγραψε: ↑
Πέμ Ιούλ 11, 2024 2:56 pm
Να υπολογιστεί το εμβαδόν της παρακάτω γραμμοσκιασμένης περιοχής.

$\displaystyle{\begin{tikzpicture} %apply color to the canvas \draw[dashed, fill=magenta] (-4, 0) -- (4, 0) -- (4, 8) -- (-4, 8) -- cycle; \draw[<->, line width=1.2pt] (4.3, 2) -- (4.3, 6); \draw (4.3, 4) node[right]{5}; %apply white color to the semicircles \draw[samples=1000, fill=white] (2, 0) arc(0:180:2cm); \draw[samples=1000, fill=white] (4, 6) arc(90:270:2cm); \draw[samples=1000, fill=white] (-4, 6) arc(90:-90:2cm); \draw[samples=1000, fill=white] (-2, 8) arc(-180:0:2cm); %fill the arcs with white color \fill[white] (-4, 2) arc(90:0:2cm) -- (-4, 2) -- (-4, 0) -- (-2, 0); \fill[white] (4, 2) arc(90:180:2cm) -- (4, 2) -- (4, 0) -- (2, 0); \fill[white] (2, 8) -- (4, 6) arc(-90:-180:2cm) -- (4, 6) -- (4, 8); \fill[white] (-4, 6) -- (-4, 6) arc(-90:0:2cm) -- (-2, 8) -- (-4, 8); %fill the interior arcs with white color \fill[white, domain=-2:0] (0, 6) arc(0:-90:2cm) -- plot(\x, {sqrt(4-(\x)^2)+4} ); \fill[white, domain=0:2] (2, 4) arc(270:180:2cm) -- plot(\x, {sqrt(4-(\x)^2)+4} ); \fill[white, domain=0:2] (2, 4) arc(90:180:2cm) -- plot(\x, {-sqrt(4-(\x)^2)+4} ); \fill[white, domain=-2:0] (-2, 4) arc(90:0:2cm) -- plot(\x, {-sqrt(4-(\x)^2)+4} ); %draw the central circle \draw (0,4) circle(2cm); %draw the arcs \draw[samples=1000] (-4, 2) arc(90:0:2cm); \draw[samples=1000] (4, 2) arc(90:180:2cm); \draw[samples=1000] (4, 6) arc(-90:-180:2cm); \draw[samples=1000] (-4, 6) arc(-90:0:2cm); %draw the arcs in the interior of the circle \draw[samples=1000] (0, 6) arc(0:-90:2cm); \draw[samples=1000] (2, 4) arc(90:180:2cm); \draw[samples=1000] (-2, 4) arc(90:0:2cm); \draw[samples=1000] (2, 4) arc(270:180:2cm); \end{tikzpicture}}$

: Ε.Γ.Π.png (14.69 KiB) Προβλήθηκε 1868 φορές

Το εμβαδόν

είναι ίσον με το εμβαδόν τετραγώνου πλευράς

μείον το εμβαδόν των τεσσάρων κυκλικών τομέων

$\displaystyle E = 25 - 4W = 25 - 4 \cdot \frac{{25\pi }}{{16}} = \frac{{25}}{4}(4 - \pi )$

Στην άσκησή μας το ζητούμενο εμβαδόν είναι $\boxed{T = 5E = \frac{{125}}{4}(4 - \pi )}$

Εμβαδόν γραμμοσκιασμένης περιοχής

Εμβαδόν γραμμοσκιασμένης περιοχής

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