Sphere and cone

Within the sphere of radius G = 33 cm inscribe the cone with the largest volume. What is that volume, and what are the dimensions of the cone?

Correct answer:

r =  31.11 cm
h =  44 cm
V =  44602.24 cm³

Step-by-step explanation:

$$\begin{gathered} G=33cm \\ V={\displaystyle \frac{1}{3}}\pi r^{2}h \\ h=G+x=33+x \\ {G}^2=x^2+r^2 \\ {r}^2=G^2-x^2 \\ V={\displaystyle \frac{1}{3}}\pi (G^2-x^2)(G+x) \\ V={\displaystyle \frac{1}{3}}\pi (G^3+G^{2}x-G{x}^2-x^3) \\ {V}^{\mathrm{\prime }}={\displaystyle \frac{1}{3}}\pi (G^2-2Gx-3x^2) \\ {V}^{\mathrm{\prime }}=0 \\ {G}^2-2Gx-3x^2=0 \\ -3x^2-66x+1089=0 \\ 3x^2+66x-1089=0 \\ a=3;b=66;c=-1089 \\ D=b^2-4ac=66^2-4\cdot 3\cdot (-1089)=17424 \\ D>0 \\ {x}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{-66\pm \sqrt{17424}}{6}} \\ {x}_{1,2}={\displaystyle \frac{-66\pm 132}{6}} \\ {x}_{1,2}=-11\pm 22 \\ {x}_1=11 \\ {x}_2=-33 \\ \text{ Factored form of the equation: } \\ 3(x-11)(x+33)=0 \\ h=G+x_1=33+11=44cm \\ r=\sqrt{G^2-x_1^2}=31.11\text{ cm } \end{gathered}$$

$h=33+11=44\text{ cm}$

$V={\displaystyle \frac{1}{3}}\pi r^{2}h=44602.24{\text{cm}}^3$

Our quadratic equation calculator calculates it.

Try another example

Showing 1 comment:

Dr Math

that's very mind blowing

3 years ago  Like

Related math problems and questions:

• Sphere in cone
  A sphere of radius 3 cm describes a cone with minimum volume. Determine cone dimensions.
• Cone
  Into rotating cone with dimensions r = 8 cm and h = 8 cm incribe cylinder with maximum volume so that the cylinder axis is perpendicular to the axis of the cone. Determine the dimensions of the cylinder.
• Paper box
  The hard rectangular paper has dimensions of 60 cm and 28 cm. The corners are cut off equal squares, and the residue was bent to form an open box. How long must beside the squares be the largest volume of the box?
• Cube in sphere
  The cube is inscribed in a sphere with a radius r = 6 cm. What percentage is the volume of the cube from the volume of the ball?
• Secret treasure
  Scouts have a tent in the shape of a regular quadrilateral pyramid with a side of the base 4 m and a height of 3 m. Find the container's radius r (and height h) so that they can hide the largest possible treasure.
• Cube in sphere
  The sphere is an inscribed cube with an edge of 8 cm. Find the sphere's radius.
• Cuboid
  Cuboid with edge a=6 cm and space diagonal u=31 cm has volume V=900 cm³. Calculate the length of the other edges.
• Body diagonal
  Calculate the length of the body diagonal of a block with dimensions: a = 20 cm, b = 30 cm, c = 15 cm.
• Ratio of edges
  The dimensions of the cuboid are in a ratio 3: 1: 2. The body diagonal has a length of 28 cm. Find the volume of a cuboid.
• Tangent spheres
  A sphere with a radius of 1 m is placed in the corner of the room. What is the largest sphere size that fits into the corner behind it? Additional info: Two spheres are placed in a corner of a room. The spheres are each tangent to the walls and floor and
• Cut and cone
  Calculate the volume of the rotation cone which lateral surface is circle arc with radius 15 cm and central angle 63 degrees.
• Truncated cone 5
  The height of a cone 7 cm and the length of side is 10 cm and the lower radius is 3cm. What could the possible answer for the upper radius of truncated cone?
• Two balls
  Two balls, one 8cm in radius and the other 6cm in radius, are placed in a cylindrical plastic container 10cm in radius. Find the volume of water necessary to cover them.
• Cone
  Calculate the volume of the rotating cone with a base radius 26.3 cm and a side 38.4 cm long.
• Maximum of volume
  The shell of the cone is formed by winding a circular section with a radius of 1. For what central angle of a given circular section will the volume of the resulting cone be maximum?
• Cube and sphere
  Cube with the surface area 150 cm² is described sphere. What is sphere surface?
• Cuboid - volume, diagonals
  The length of the one base edge of cuboid a is 3 cm. Body diagonal is ut=13 cm and diagonal of cuboid's baseis u1=5 cm. What is the volume of the cuboid?