Answer
(a) $0.5625$
(b) $0.4219$
Work Step by Step
We can write the general equation of a sphere:
$(x-a)^2+(y-b)^2+(z-c)^2 = r^2$
where $(a,b,c)$ is the center of the sphere and $r$ is the radius
The equation of the first sphere is: $(x-1)^2+(y+2)^2+(z-4)^2 = 36$
The radius of the first sphere is $r_1 = 6$
The equation of the second sphere is: $x^2+y^2+z^2 = 64$
The radius of the second sphere is $r_2 = 8$
(a) We can find the surface area of the first sphere:
$A_1 = 4~\pi~r_1^2$
$A_1 = (4~\pi)~(6)^2$
$A_1 = 144~\pi$
We can find the surface area of the second sphere:
$A_2 = 4~\pi~r_2^2$
$A_2 = (4~\pi)~(8)^2$
$A_2 = 256~\pi$
We can find the ratio of the surface areas:
$\frac{A_1}{A_2} = \frac{144~\pi}{256~\pi} = 0.5625$
(b) We can find the volume of the first sphere:
$V_1 = \frac{4}{3}~\pi~r_1^3$
$V_1 = \frac{4}{3}~\pi~(6)^3$
$V_1 = 288~\pi$
We can find the volume of the second sphere:
$V_2 = \frac{4}{3}~\pi~r_2^3$
$V_2 = \frac{4}{3}~\pi~(8)^3$
$V_2 = 682.7~\pi$
We can find the ratio of the volumes:
$\frac{V_1}{V_2} = \frac{288~\pi}{682.7~\pi} = 0.4219$
