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Podręcznik : Matematyka z pLusem . klasa 2 gimnazjum ! [książka z niebieską okładką . !
Zad. 9 str. 64
a.) Oblicz pole koła o obwodzie :
16π cm π m 0,12π mm 2/3π km 4dm
b.) Oblicz obwód koła o polu :
16π cm² 4/9π m² 0,25π mm² 2π dm² 3km²

Odpowiedź :

Anıaviz
[tex]P=\pi r^2\\ Obw=2\pi r\\ \\ [/tex]

[tex]2\pi r=16\pi \ cm \ \ /:2\pi\\ r=8cm\\ P=\pi*8^2=64\pi \ cm^2\\ \\\\ 2\pi r=\pi \ m \ \ /:2\pi\\ r=\frac{1}{2}m\\ P=\pi*(\frac{1}{2})^2=\frac{1}{4}\pi m^2\\\\\\ 2\pi r=0,12\pi mm\ \ /:2\pi\\ r=0,06mm\\ P=\pi*(0,06)^2=0,0036\pi mm^2\\\\\\ 2\pi r=\frac{2}{3}\pi km \ \ /:2\pi\\ r=\frac{1}{3} km\\ P=\pi*(\frac{1}{3})^2=\frac{1}{9}\pi km^2\\\\\\ 2\pi r=4dm \ \ /:2\pi\\ r=\frac{2}{\pi}dm\\ P=\pi*(\frac{2}{\pi})^2=\pi*\frac{4}{\pi^2}=\frac{4}{\pi}dm^2\\[/tex]


[tex]\pi r^2=16\pi cm^2 \ \ /: \pi\\ r^2=16cm^2\\ r=4cm\\ Obw=2*4\pi =8\pi cm\\\\ \\ \pi r^2=\frac{4}{9}\pi m^2 \ \ /: \pi\\ r^2=\frac{4}{9}m^2\\ r=\frac{2}{3}m\\ Obw=2*\frac{2}{3}\pi =\frac{4}{3}\pi m \\ \\ \\ \pi r^2=0,25\pi mm^2 \ \ /: \pi\\ r^2=0,25mm^2\\ r=0,5mm\\ Obw=2*0,5\pi =\pi mm\\ \\ \\ \pi r^2=2\pi dm^2 \ \ /: \pi\\ r^2=2dm^2\\ r=\sqrt{2}dm\\ Obw=2*\sqrt{2}\pi =2\sqrt{2}\pi dm \\\\ \\ [/tex]

[tex]\pi r^2=3 \ km^2 \ \ /: \pi\\ r^2=\frac{3}{\pi} \ km^2\\ r=\frac{\sqrt{3}}{\sqrt{\pi}} \ km\\ Obw=2*\frac{\sqrt{3}}{\sqrt{\pi}}\pi =2\sqrt{3\pi} \ km \\ \\[/tex]
Aniabaranekviz
a)

[tex]P= \pi r^{2} Obw=2 \pi r [/tex]

Aby policzyć pole należy przekształcić wzór obwodu tak, aby wyznaczyć r:
[tex]Obw=2 \pi r r= Obw : (2 \pi ) [/tex]

[tex] Obw = 16 \pi [cm] r=(16 \pi )/(2 \pi )=8 P= \pi* 8^2= 64 \pi [cm^2] [/tex]

[tex]Obw = \pi [m] r=(\pi )/(2 \pi )=1/2 P= \pi *(\frac{1}{2} )^2= \frac{1}{4} \pi [m^2][/tex]

[tex]Obw = 0,12 \pi [mm] r=(0,12 \pi )/(2 \pi )=0,06 P= \pi *0,06^2= 0,0036 \pi [mm^2][/tex]

[tex]Obw = 4 [dm] r=(4 )/(2 \pi )=2/ \pi P= \pi *(2/ \pi)^2= 4 / \pi [dm^2][/tex]

[tex]Obw = \frac{2}{3} \pi [km] r = (2/3 \pi):(2 \pi )= \frac{1}{3} P= \pi *( \frac{1}{3})^2= \frac{1}{9} \pi [km^2][/tex]

b) Wyznaczam ze wzoru na pole promień okręgu r:
[tex]P= \pi r^{2} r= \sqrt{ \frac{P}{ \pi } } [/tex]


[tex]P= 16 \pi [cm^2] r= \sqrt{ \frac{16 \pi }{ \pi } } = \sqrt{ 16 } = 4 Obw = 2 \pi r=2 \pi *4 = 8 \pi [cm][/tex]


[tex]P= \frac{4}{9} \pi [m^2] r= \sqrt{ \frac{ \frac{4}{9} \pi }{ \pi } } = \sqrt{ \frac{4}{9} } = \frac{2}{3} Obw = 2 \pi r=2 \pi* \frac{2}{3} = \frac{4}{3} \pi [m][/tex]

[tex]P= \frac{1}{4} \pi [mm^2] \\ r= \sqrt{ \frac{ \frac{1}{4} \pi }{ \pi } } = \sqrt{ \frac{1}{4} } = \frac{1}{2}\\ Obw = 2 \pi r=2 \pi* \frac{1}{2} = \pi [mm][/tex]

[tex]P= 3 [km^2] \\ r= \sqrt{ \frac{3 }{ \pi } }\\ Obw = 2 \pi r=2 \pi* \sqrt{ \frac{3 }{ \pi } }=2 \sqrt{3 \pi } [km][/tex]