+823 A cone contains two spheres, as shown below. If the radii of the spheres are 1 and $4,$ then find the volume of the cone.
+130042 I oriented the spheres like this
We can set up similar triangles AEC and BDC
We have that
AE/ AC = BD / BC
FC = x
AE= 4
AC = 6 + x
BD = 1
BC = 1 + x
So
4/ (6 +x) = 1/ (1 + x)
4(1 + x) = 6 +x
4 + 4x = 6 + x
3x = 2
x = 2/3
C = ( 4 + 2 + 2/3 , 0) = (20/3 , 0)
HC = 6+ 2 + x = 6 + 2 + 2/3 = 26/3 = height of cone
We can find DC as sqrt ( (5/3)^2 -1 ) =sqrt ( 25/9 - 9/9) = sqrt (16/9) = 4/3
The equation of the tangent line to both circles through D has the slope = -BD/DC = -1/ ( 4/3) = -3/4
And the equation of this line is
y = (-3/4)(x -20/3)
The intersection of this line with the line x =- 4 gives us the radius of the cone =
y = (-3/4)(-4 -20/3) = (-3/4)(-32/3) = 8 = GH
So the volume of the cone = (1/3)pi GH^2 * HC = (1/3)pi * 8^2 * (26/3) = (1664/9) pi
