Sphere and cone

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

Correct result:

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

Solution:

G=33 cm V=13πr2h h=G+x=33+x G2=x2+r2 r2=G2−x2 V=13π(G2−x2)(G+x) V=13π(G3+G2x−Gx2−x3)  V′=13π(G2−2Gx−3x2) V′=0 G2−2Gx−3x2=0 −3x2−66x+1089=0 3x2+66x−1089=0  a=3;b=66;c=−1089 D=b2−4ac=662−4⋅3⋅(−1089)=17424 D>0  x1,2=−b±D2a=−66±174246 x1,2=−66±1326 x1,2=−11±22 x1=11 x2=−33   Factored form of the equation:  3(x−11)(x+33)=0   h=G+x1=33+11=44 cm r=G2−x12=31.11 cm G = 33 \ cm \ \\ V = \dfrac13 \pi r^2 h \ \\ h = G + x = 33 +x \ \\ G^2 = x^2+r^2 \ \\ r^2 = G^2 - x^2 \ \\ V = \dfrac13 \pi (G^2 - x^2)(G+x) \ \\ V = \dfrac13 \pi (G^3 + G^2x - Gx^2 - x^3) \ \\ \ \\ V' = \dfrac13 \pi (G^2 - 2Gx - 3x^2) \ \\ V'=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} = \dfrac{ -b \pm \sqrt{ D } }{ 2a } = \dfrac{ -66 \pm \sqrt{ 17424 } }{ 6 } \ \\ x_{1,2} = \dfrac{ -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 = 44 \ cm \ \\ r = \sqrt{ G^2 - x_1^2 } = 31.11 \ \text{cm} \ \\
h=33+11=44 cm
V=13πr2h=44602.24 cm3

Checkout calculation with our calculator of quadratic equations.


Try another example


We would be very happy if you find an error in the example, spelling mistakes, or inaccuracies, and please send it to us. We thank you!






Showing 1 comment:
#
Dr Math
that's very mind blowing

avatar









Tips to related online calculators
Looking for help with calculating roots of a quadratic equation?
Pythagorean theorem is the base for the right triangle calculator.
See also our trigonometric triangle calculator.

You need to know the following knowledge to solve this word math problem:


 
We encourage you to watch this tutorial video on this math problem: video1   video2   video3

Next similar math problems:

  • Cube in sphere
    sphere4 The sphere is inscribed cube with edge 8 cm. Find the radius of the sphere.
  • Sphere from tree points
    sphere2_1 Equation of sphere with three point (a,0,0), (0, a,0), (0,0, a) and center lies on plane x+y+z=a
  • Cone
    diag22 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.
  • RT and circles
    r_triangle Solve right triangle if the radius of inscribed circle is r=9 and radius of circumscribed circle is R=23.
  • Triangle ABC
    lalala In a triangle ABC with the side BC of length 2 cm The middle point of AB. Points L and M split AC side into three equal lines. KLM is isosceles triangle with a right angle at the point K. Determine the lengths of the sides AB, AC triangle ABC.
  • Euclid2
    euclid In right triangle ABC with right angle at C is given side a=27 and height v=12. Calculate the perimeter of the triangle.
  • Combinations
    math_2 From how many elements we can create 990 combinations 2nd class without repeating?
  • Equation
    calculator_2 Equation ? has one root x1 = 8. Determine the coefficient b and the second root x2.
  • Quadratic equation
    kvadrat_2 Find the roots of the quadratic equation: 3x2-4x + (-4) = 0.
  • Roots
    parabola Determine the quadratic equation absolute coefficient q, that the equation has a real double root and the root x calculate: ?
  • Discriminant
    Quadratic_equation_discriminant Determine the discriminant of the equation: ?
  • Catheti
    pyt_theorem The hypotenuse of a right triangle is 41 and the sum of legs is 49. Calculate the length of its legs.
  • ABS CN
    complex_num Calculate the absolute value of complex number -15-29i.
  • RT triangle and height
    345 Calculate the remaining sides of the right triangle if we know side b = 4 cm long and height to side c h = 2.4 cm.
  • Vector 7
    vectors_sum0_1 Given vector OA(12,16) and vector OB(4,1). Find vector AB and vector |A|.
  • RTriangle 17
    rt The hypotenuse of a right triangle is 17 cm. If you decrease both two legs by 3 cm you will reduce the hypotenuse by 4 cm. Determine the length of this legs.
  • Isosceles triangle
    triangle2_3 The leg of the isosceles triangle is 5 dm, its height is 20 cm longer than the base. Calculate base length z.