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.

Correct result:

r =  2.67 cm
h =  5.33 cm

Solution:

V=13πr2h x/R=2r/H  r=H2Rx=828x h=2(Rx)=2(8x)  V=13π(828x)22(8x)  V=π82682(8x2x3)  V=π82682(28x3x2) V=0 28x3x2=0 x1=0 283x=0  16=3x  x=5.333 cm r=8285.333=2.67 cm V = \dfrac13 \pi r^2 h \ \\ x/R = 2r/H \ \\ \ \\ r = \dfrac{H }{ 2R } x = \dfrac{ 8 }{ 2 \cdot 8 } x \ \\ h = 2(R-x) = 2(8-x) \ \\ \ \\ V = \dfrac13 \pi (\dfrac{ 8 }{ 2 \cdot 8 } x )^2 \cdot 2(8-x) \ \\ \ \\ V = \dfrac{ \pi 8^2}{ 6 \cdot 8^2 } ( 8 x^2- x^3) \ \\ \ \\ V` = \dfrac{ \pi 8^2}{ 6 \cdot 8^2 } ( 2\cdot 8 x - 3x^2) \ \\ V` = 0 \ \\ 2\cdot 8 x - 3x^2 = 0 \ \\ x_1 = 0 \ \\ 2\cdot 8 - 3x = 0 \ \\ \ \\ 16 = 3x \ \\ \ \\ x = 5.333 \ cm \ \\ r = \dfrac{ 8 }{ 2 \cdot 8 } \cdot 5.333 = 2.67 \ \text{cm} \ \\
h=2(85.333)=5.33 cmh = 2(8-5.333) = 5.33 \ \text{cm}



Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!


Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Showing 0 comments:
1st comment
Be the first to comment!
avatar




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

Next similar math problems:

  • Jar
    sklenice From the cylinder shaped jar after tilting spilled water so that the bottom of the jar reaches the water level accurately into half of the base. Height of jar h = 7 cm and a jar diameter D is 12 cm. How to calculate how much water remains in the jar?
  • Linsys2
    linear_eq_3 Solve two equations with two unknowns: 400x+120y=147.2 350x+200y=144
  • Cylindrical container
    valec2_6 An open-topped cylindrical container has a volume of V = 3140 cm3. Find the cylinder dimensions (radius of base r, height v) so that the least material is needed to form the container.
  • Paper box
    box 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 be side of the squares to be the largest volume of the box?
  • Shopping malls
    tv The chain of department stores plans to invest up to 24,000 euros in television advertising. All commercials will be placed on a television station where the broadcast of a 30-second spot costs EUR 1,000 and is watched by 14,000 potential customers, durin
  • Curve and line
    parabol The equation of a curve C is y=2x² -8x+9 and the equation of a line L is x+ y=3 (1) Find the x co-ordinates of the points of intersection of L and C. (2) Show that one of these points is also the stationary point of C?
  • Statue
    michelangelo On the pedestal high 4 m is statue 2.7 m high. At what distance from the statue must observer stand to see it in maximum viewing angle? Distance from the eye of the observer from the ground is 1.7 m.
  • Inverse matrix
    matrix_3 Find how many times is the larger determinant is the matrix A, which equals 9 as the determinant of its inverse matrix.
  • The determinant
    matrix_13 The determinant of the unit matrix equals 7. Check how many rows the A matrix contains.
  • Minimum of sum
    derive_1 Find a positive number that the sum of the number and its inverted value was minimal.
  • Derivation
    fx Exists a function whose derivation is the same function?
  • Calculation of CN
    color_combinations Calculate: ?
  • Equation
    calculator_2 Equation ? has one root x1 = 8. Determine the coefficient b and the second root x2.
  • Complex number coordinates
    complex_numbers Which coordinates show the location of -2+3i
  • Sequence 3
    75 Write the first 5 members of an arithmetic sequence: a4=-35, a11=-105.
  • Researchers
    house Researchers ask 200 families whether or not they were the homeowner and how many cars they had. Their response was homeowner: 14 no car or one car, two or more cars 86, not homeowner: 38 no car or one car, two or more cars 62. What percent of the families
  • Seats
    divadlo_2 Seats in the sport hall are organized so that each subsequent row has five more seats. First has 10 seats. How many seats are: a) in the eighth row b) in the eighteenth row