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?

Correct result:

A =  293.9388 °

Solution:

$$\begin{gathered} s=1 \\ V={\displaystyle \frac{1}{3}}\cdot \pi \cdot r^2\cdot h \\ h=\sqrt{s^2-r^2} \\ V={\displaystyle \frac{1}{3}}\cdot \pi \cdot r^2\cdot \sqrt{s^2-r^2} \\ V={\displaystyle \frac{1}{3}}\cdot \pi \cdot r^2\cdot \sqrt{1-r^2} \\ {V}^{\mathrm{\prime }}=0 \\ {V}^{\mathrm{\prime }}={\displaystyle \frac{dV}{dr}}={\displaystyle \frac{\pi r(2-3r^2)}{3\sqrt{1-r^2}}} \\ r>0;r\mathrm{\ne }1 \\ 2-3r^2=0 \\ r=\sqrt{2\mathrm{/}3}\doteq 0.8165 \\ {o}_1=2\pi \cdot r=2\cdot 3.1416\cdot 0.8165\doteq 5.1302 \\ {o}_2=2\pi \cdot s=2\cdot 3.1416\cdot 1\doteq 6.2832 \\ A=360\cdot {\displaystyle \frac{o_1}{o_2}}=360\cdot {\displaystyle \frac{5.1302}{6.2832}}=120\sqrt{6}=293.9388^{\circ }=293^{\circ }56^{\mathrm{\prime }}20\mathrm{"} \end{gathered}$$

Try another example

Showing 0 comments:

Next similar math problems:

• Cone A2V
  Surface of cone in the plane is a circular arc with central angle of 126° and area 415 cm². Calculate the volume of a cone.
• 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.
• Circular pool
  The base of the pool is a circle with a radius r = 10 m, excluding a circular segment that determines the chord length 10 meters. The pool depth is h = 2m. How many hectoliters of water can fit into the pool?
• Hexagon rotation
  A regular hexagon of side 6 cm is rotated through 60° along a line passing through its longest diagonal. What is the volume of the figure thus generated?
• 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?
• Circular segment
  Calculate the area S of the circular segment and the length of the circular arc l. The height of the circular segment is 2 cm and the angle α = 60°. Help formula: S = 1/2 r2. (Β-sinβ)
• Circular sector
  I have a circular sector with a length 15 cm with an unknown central angle. It is inscribed by a circle with radius 5 cm. What is the central angle alpha in the circular sector?
• 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. Determine the radius r (and height h) of the container so that they can hide the largest possible treasure.
• Volume of the cone
  Calculate the volume of the cone if the content of its base is 78.5 cm² and the content of the shell is 219.8 cm².
• Circle arc
  Circle segment has a circumference of 135.26 dm and 2096.58 dm² area. Calculate the radius of the circle and size of central angle.
• Surface of the cone
  Calculate the surface of the cone if its height is 8 cm and the volume is 301.44 cm³.
• Axial section
  Axial section of the cylinder has a diagonal 40 cm. The size of the shell and the base surface are in the ratio 3:2. Calculate the volume and surface area of this cylinder.
• Cylinder horizontally
  The cylinder with a diameter of 3 m and a height/length of 15 m is laid horizontally. Water is poured into it, reaching a height of 60 cm below the axis of the cylinder. How many hectoliters of water is in the cylinder?
• Sphere parts, segment
  A sphere with a diameter of 20.6 cm, the cut is a circle with a diameter of 16.2 cm. .What are the volume of the segment and the surface of the segment?
• Circle section
  Equilateral triangle with side 33 is inscribed circle section whose center is in one of the vertices of the triangle and the arc touches the opposite side. Calculate: a) the length of the arc b) the ratio betewwn the circumference to the circle sector and
• Cross-sections of a cone
  Cone with base radius 16 cm and height 11 cm divide by parallel planes to base into three bodies. The planes divide the height of the cone into three equal parts. Determine the volume ratio of the maximum and minimum of the resulting body.
• Circle sector
  Circular sector with a central angle 80 ° has area 1257 cm². Calculate its radius r.