Respuesta :
we know that
For a spherical planet of radius r, the volume V and the surface area SA is equal to
[tex] V=\frac{4}{3} *\pi *r^{3} \\ \\ SA=4*\pi *r^{2} [/tex]
The
ratio of these two quantities may be written as
[tex] SAV =\frac{(4*\pi*r^{2})}{(\frac{4}{3}*\pi*r^{3})} \\ \\ SAV =\frac{3}{r} [/tex]
we know
[tex] rMoon=1,738Km\\ rMars=3,397 Km [/tex]
[tex] \frac{SAV Moon}{SAV Mars} =\frac{3}{rMoon} *\frac{rMars}{3} \\ \\ \frac{SAV Moon}{SAV Mars} =\frac{rMars}{rMoon} \\ \\ \frac{SAV Moon}{SAV Mars} =\frac{3,397}{1,738} \\ \\ \frac{SAV Moon}{SAV Mars} =1.9545 [/tex]
therefore
the answer is
[tex] 1.9 [/tex]
The ratio of surface area to volume of the moon is [tex]\boxed{\bf 1.9\text{\bf\ times}}[/tex] of the ratio of surface area to volume of the mars.
Further explanation:
The volume of the spherical shape can be calculated as,
[tex]\boxed{\text{Volume of sphere}=\dfrac{4}{3}\pi r^{3}}[/tex]
The surface area of the spherical shape can be calculated as,
[tex]\boxed{\text{Surface atrea of sphere}=4\pi r^{2}}[/tex]
Here, [tex]r[/tex] is the radius of the sphere.
Calculation:
Step 1:
First we will calculate the ratio of surface area of the sphere to the volume of the sphere.
The ratio of these quantities can be calculated as,
[tex]\begin{aligned}\dfrac{\text{Surface area}}{\text{Volume}}&=\dfrac{4\pi r^{2}}{\dfrac{4}{3}\pi r^{3}}\\&=\dfrac{3}{r}\end{aligned}[/tex]
Therefore, the ratio of the surface area to the volume of the sphere is [tex]\frac{3}{r}[/tex].
Step 2:
Second we will calculate the ratio of surface area of the moon to the volume of the moon.
The radius of the moon is [tex]1738\text{ km}[/tex].
Substitute [tex]r=1738[/tex] in the ratio of the surface area to the volume of the sphere.
[tex]\begin{aligned}\dfrac{\text{Surface area of the moon}}{\text{Volume of the moon}}&=\dfrac{3}{r}\\&=\dfrac{3}{1738}\end{aligned}[/tex]
Step 3:
Third we will calculate the ratio of surface area of the mars to the volume of the mars.
The radius of the mars is [tex]3397\text{ km}[/tex].
Substitute [tex]r=3397[/tex] in the ratio of the surface area to the volume of the sphere.
[tex]\begin{aligned}\dfrac{\text{Surface area of the mars}}{\text{Volume of the mars}}&=\dfrac{3}{r}\\&=\dfrac{3}{3397}\end{aligned}[/tex]
Step 4:
Now compare the ratios of the surface area to volume of moon with ratios of the surface area to volume of mars.
Divide the ratios of the moon by the ratio of the mars to compare.
[tex]\begin{aligned}\dfrac{\text{ratio for moon}}{\text{ratio for mars}}&=\dfrac{1738}{3}\times\dfrac{3}{3397}\\&=\dfrac{1738}{3397}\\&=1.9\end{aligned}[/tex]
Clearly we can see that the ratio surface area to volume of the moon is [tex]\boxed{\bf 1.9\text{\bf\ times}}[/tex] of the ratio surface area to volume of the mars.
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Answer details:
Grade: Middle school
Subject: Mathematics
Chapter: Mensuration
Keywords: Mensuration, volume, surface area, sphere, ratio, moon, mars, comparision, spherical shape, curved surface area, radius of the sphere.
