SOLUTION: The ratio of the altitude of a zone to the diameter of the sphere on which it is 1:5. The area of the zone is 80 pi. Find the area of the sphere.

Algebra ->  Numeric Fractions Calculators, Lesson and Practice -> SOLUTION: The ratio of the altitude of a zone to the diameter of the sphere on which it is 1:5. The area of the zone is 80 pi. Find the area of the sphere.      Log On


   

Question 1203307: The ratio of the altitude of a zone to the diameter of the sphere on which it is 1:5. The area of the zone is 80 pi. Find the area of the sphere.
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

Surface area of a sphere:
A=4%2Api%2Ar%5E2

let altitude be h and diameter d
given ratio is 1%3A5

so, h%2Fd=1%2F5

h=d%2F5
r=d%2F2+=> h=%28d%2F2%29%2F5=d%2F10

The area A of a spherical zone can be calculated using the formula

A=2%C3%8F%E2%82%ACrh+
where h+is the height of the spherical layer and r is the radius of the sphere.

if the area of the zone is 80pi, we have
80pi=2pi%2Arh...simplify
40=rh.....substitute h
40=r%28r%2F10%29
400=r%5E2
r=sqrt%28400%29
r=20

then, surface area of a sphere is:
A=4%2Api%2A20%5E2
A=4%2Api%2A400
A=1600pi



Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

        The solution by @MathLover1 is incorrect.
        I came to bring you a correct solution.


Surface area of a sphere:
A=4%2Api%2Ar%5E2

let altitude be h and diameter d
given ratio is 1%3A5

so, h%2Fd=1%2F5

h=d%2F5
r=d%2F2+   -->   h=%282r%29%2F5         <<<---===   it is where  @MathLover1  made her mistake.

The area A of a spherical zone can be calculated using the formula

A=2%2Api%2Arh+
where h+is the height of the spherical zone and r is the radius of the sphere.

if the area of the spherical zone is 80pi, we have
80pi=2%2Api%2Arh...simplify
40=rh.....substitute h
40=r%2A%282r%29%2F5%29
100=r%5E2
r=sqrt%28100%29
r=10

then, surface area of a sphere is:
A=4%2Api%2A10%5E2
A=4%2Api%2A100
A=400%2Api         ANSWER

Solved   (correctly).