Respuesta :
Answer:
The volume of the sphere is [tex]V\approx101.212 \:in^3[/tex].
Step-by-step explanation:
A sphere is a 3-D figure in which all of the points in a plane are the same distance from a given point, the center of the sphere.
A sphere with radius r has a volume of
[tex]V=\frac{4}{3} \pi r^3[/tex]
and a surface area of
[tex]S=4\pi r^2[/tex]
To find the volume of the sphere we use the fact that the surface area of the sphere is 105 [tex]in^2[/tex] and we use it to find the radius.
[tex]105=4\pi r^2\\\\4\left(\pi \right)r^2=105\\\\r^2=\frac{105}{4\pi }\\\\\mathrm{For\:}x^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}x=\sqrt{f\left(a\right)},\:\:-\sqrt{f\left(a\right)}\\\\r=\sqrt{\frac{105}{4\pi }},\:r=-\sqrt{\frac{105}{4\pi }}[/tex]
The radius cannot be negative. Therefore,
[tex]r=\sqrt{\frac{105}{4\pi }}=\frac{\sqrt{105}\sqrt{\pi }}{2\pi }\approx 2.891 \:in[/tex]
Now, that we know the radius we can find the volume
[tex]V=\frac{4}{3} \pi (2.891)^3=\frac{96.65053\dots \pi }{3}=\frac{303.63661\dots }{3}\approx101.212 \:in^3[/tex]
