Cylindrical container

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.

Correct result:

r =  9.9983 cm
v =  9.9983 cm

Solution:

$$\begin{gathered} V=3140{\text{cm}}^3 \\ V=\pi r^{2}v \\ v=V\mathrm{/}(\pi r^2) \\ S=\pi r^2+2\pi rv \\ S=\pi r^2+2\pi rV\mathrm{/}(\pi r^2) \\ S=\pi r^2+2\cdot 3140\mathrm{/}r \\ {S}^{\mathrm{\prime }}=2\pi r-6280\mathrm{/}{r}^2 \\ {S}^{\mathrm{\prime }}=0 \\ 2\pi r=6280\mathrm{/}{r}^2 \\ 2\pi r^3=6280 \\ r=\sqrt[3]{6280\mathrm{/}(2\pi )}=\sqrt[3]{6280\mathrm{/}(2\cdot 3.1416)}\text{ cm}=9.9983\text{ cm} \end{gathered}$$

$v=V\mathrm{/}(\pi \cdot r^2)=3140\mathrm{/}(3.1416\cdot 9.9983^2)=9.9983\text{ cm}$

Try another example

Showing 0 comments:

Related math problems and questions:

• 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
  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?
• Cylinder surface, volume
  The area of the cylinder surface and the cylinder jacket are in the ratio 3: 5. The height of the cylinder is 5 cm shorter than the radius of the base. Calculate surface area and volume of the cylinder.
• The cylindrical container
  The cylindrical container has a base area of 300 cm³ and a height of 10 cm. It is 90% filled with water. We gradually insert metal balls into the water, each with a volume of 20 cm³. After inserting how many balls for the first time does water flow over
• 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.
• Sphere in cone
  A sphere of radius 3 cm describes a cone with minimum volume. Determine cone dimensions.
• 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.
• Rectangle pool
  Determine dimensions of open pool with a square bottom with a capacity 32 m³ to have painted/bricked walls with least amount of material.
• The cylindrical container
  The container has a cylindrical shape the base diameter 0.8 meters has a content area of the base is equal to the content area of the shell. How many full liters of water can be poured maximally into the container?
• Axial section
  The axial section of the cylinder has a diagonal 31 cm long, and we know that the area of the side and the base area is in ratio 3:2. Calculate the height and radius of the cylinder base.
• Minimum surface
  Find the length, breadth, and height of the cuboid-shaped box with a minimum surface area, into which 50 cuboid-shaped blocks, each with length, breadth, and height equal to 4 cm, 3 cm, and 2 cm respectively, can be packed.
• 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.
• Shell area cy
  The cylinder has a shell content of 300 cm square, while the height of the cylinder is 12 cm. Calculate the volume of this cylinder.
• 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?
• Cylinder surface area
  Volume of a cylinder whose height is equal to the radius of the base is 678.5 dm³. Calculate its surface area.
• Alien ship
  The alien ship has the shape of a sphere with a radius of r = 3000m, and its crew needs the ship to carry the collected research material in a cuboid box with a square base. Determine the length of the base and (and height h) so that the box has the large
• Cylinder
  Calculate the dimensions of rotating cylindrical containers with volume 2 l if the container's height is equal to the base's diameter.