Shown below is a sphere, cone and cube. The surface area of the sphere is equal to the sum the surface areas of the cone and cube. so what is Y's number?

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Answer 1 · 2018-03-15T09:57:52

#color(blue)(y=r=sqrt(9+96/pi)~~6.2895)#

First calculate the surface areas of the cone and cube.

The surface area of the cube:

We have 6 faces each having a surface area of:

#8*8=64 "cm"^2#

Total surface area:

#6*64=384"cm"^2#

The surface area of a cone is given by:

#bb(A=pir(r+sqrt(h^2+r^2)))#

#h# is the height of the cone. We are given the length of the slope, so we will need to find #h#.

We can do this by using Pythagoras's theorem.

If the slope is the hypotenuse, and the radius is one of the sides of a right triangle, then the missing side #h# is:

#h=sqrt((9)^2-(3)^2)=sqrt(72)=6sqrt(2)#

Putting known values into the formula:

#A=pi(3)((3)+sqrt((6sqrt(2))^2+(3)^2))#

#A=pi(3)((3)+sqrt((72+9)))#

#A=pi(3)(12)=36pi#

Surface area of sphere is the sum of the surface area of the cube and the cone:

#(36pi+384) "cm"^2#

Surface area of a sphere is given by:

#bb(A=4pir^2)#

#:.#

#4pir^2=36pi+384#

We need to find #r# which will be our #y# value:

#4pir^2=36pi+384#

Divide by #4pi#:

#r^2=(36pi)/(4pi)+384/(4pi)#

#r^2=9+96/pi#

Taking square root:

#r=sqrt(9+96/pi)#

You can leave it in this form for an exact answer, or its approximate value of:

#color(blue)(y=r=sqrt(9+96/pi)~~6.2895)#