Lateral surface area

The ratio of the area of the base of the rotary cone to its lateral surface area is 3: 5. Calculate the surface and volume of the cone, if its height v = 4 cm.

Correct answer:

S =  75.3982 cm²
V =  37.6991 cm³

Step-by-step explanation:

$$\begin{gathered} v=4\text{ cm } \\ {S}_1:S_2=3:5 \\ {S}_1=\pi r^2 \\ {S}_2=\pi rs \\ 3\mathrm{/}5={\displaystyle \frac{\pi r^2}{\pi rs}} \\ 3\mathrm{/}5=r\mathrm{/}s \\ {s}^2=r^2+v^2 \\ {s}^2=(3\mathrm{/}5s{)}^2+v^2 \\ s=v\mathrm{/}\sqrt{1-(3\mathrm{/}5{)}^2}=4\mathrm{/}\sqrt{1-(3\mathrm{/}5{)}^2}=5\text{ cm } \\ r=3\mathrm{/}5\cdot s=3\mathrm{/}5\cdot 5=3\text{ cm } \\ {S}_1=\pi \cdot r^2=3.1416\cdot 3^2\doteq 28.2743{\text{cm}}^2 \\ {S}_2=\pi \cdot r\cdot s=3.1416\cdot 3\cdot 5\doteq 47.1239{\text{cm}}^2 \\ k=S_1\mathrm{/}{S}_2=28.2743\mathrm{/}47.1239={\displaystyle \frac{3}{5}}=0.6 \\ S=S_1+S_2=28.2743+47.1239=75.3982{\text{cm}}^2 \end{gathered}$$

$V={\displaystyle \frac{1}{3}}\cdot S_1\cdot v={\displaystyle \frac{1}{3}}\cdot 28.2743\cdot 4=37.6991{\text{cm}}^3$

Try another example

Related math problems and questions:

• Cone and the ratio
  The rotational cone has a height 43 cm, and the ratio of the base surface to lateral surface is 5: 7. Calculate the surface of the base and the lateral surface.
• Frustrum - volume, area
  Calculate the surface and volume of the truncated cone, the radius of the smaller figure is 4 cm, the height of the cone is 4 cm and the side of the truncated cone is 5 cm.
• The diagram 2
  The diagram shows a cone with slant height 10.5cm. If the curved surface area of the cone is 115.5 cm². Calculate correct to 3 significant figures: *Base Radius *Height *Volume of the cone
• Axial section of the cone
  The axial section of the cone is an isosceles triangle in which the ratio of cone diameter to cone side is 2: 3. Calculate its volume if you know its area is 314 cm square.
• Base of prism
  The base of the perpendicular prism is a rectangular triangle whose legs length are at a 3: 4 ratio. The height of the prism is 2cm smaller than the larger base leg. Determine the volume of the prism if its surface is 468 cm².
• Slant height
  The cone's slant height is 5cm, and the radius of its base is 3cm, find the volume of the cone.
• Cone container
  Rotary cone-shaped container has a volume 1000 cubic cm and a height 12 cm. Calculate how much metal we need for making this package.
• Cone - from volume surface area
  The volume of the rotating cone is 1,018.87 dm³, and its height is 120 cm. What is the surface area of the cone?
• Surface of the cone
  Calculate the surface of the cone if its height is 8 cm and the volume is 301.44 cm³.
• Cone side
  Calculate the volume and area of the cone whose height is 10 cm and the axial section of the cone has an angle of 30 degrees between height and the cone side.
• Rotary bodies
  The rotating cone and the rotary cylinder have the same volume of 180 cm³ and the same height v = 15 cm. Which of these two bodies has a larger surface area?
• Rotary cone
  The volume of the rotation of the cone is 472 cm³, and the angle between the side of the cone and the base angle is 70°. Calculate the lateral surface area of this cone.
• Truncated cone
  Find the volume and surface area of the truncated cone if r1 = 12 cm, r2 = 5 cm and side s = 10 cm.
• Triangular prism - regular
  The regular triangular prism is 7 cm high. Its base is an equilateral triangle whose height is 3 cm. Calculate the surface and volume of this prism.
• Ratio-cuboid
  The lengths of the edges of the cuboid are in the ratio 2: 3: 6. Its body diagonal is 14 cm long. Calculate the volume and surface area of the cuboid.
• Angle of deviation
  The surface of the rotating cone is 30 cm² (with circle base), its surface area is 20 cm². Calculate the deviation of the side of this cone from the plane of the base.
• Rotating cone
  Find the rotating cone's surface and volume if its side is 150 mm long and the circumference of the base is 43.96 cm.