Above Earth

To what height must a boy be raised above the earth to see one-fifth of its surface.

Correct answer:

h =  4252 km

Step-by-step explanation:

$$\begin{gathered} R=6378\text{ km } \\ {S}_1=4\pi \cdot R^2=4\cdot 3.1416\cdot 6378^2\doteq 511185932.522{\text{km}}^2 \\ S=1\mathrm{/}5\cdot S_1=1\mathrm{/}5\cdot 511185932.522=102237186.505{\text{km}}^2 \\ S=2\pi Rv \\ v=S\mathrm{/}(2\pi \cdot R)=102237186.505\mathrm{/}(2\cdot 3.1416\cdot 6378)={\displaystyle \frac{12756}{5}}=2551.2 \\ y=R-v=6378-2551.2={\displaystyle \frac{19134}{5}}=3826.8 \\ x=\sqrt{R^2-y^2}=\sqrt{6378^2-3826.8^2}={\displaystyle \frac{25512}{5}}=5102.4 \\ k=\text{cotan}\alpha =x\mathrm{/}y=(h+v)\mathrm{/}x \\ x\mathrm{/}y=(h+v)\mathrm{/}x \\ h+v=x^2\mathrm{/}y \\ h=x^2\mathrm{/}y-v=5102.4^2\mathrm{/}3826.8-2551.2=4252\text{ km} \end{gathered}$$

Try another example

Related math problems and questions:

• Spherical cap 4
  What is the surface area of a spherical cap, the base diameter 20 m, height 2.5 m? Calculate using formula.
• Pilot
  How high is the airplane's pilot to see 0.001 of Earth's surface?
• Spherical cap
  From the sphere of radius 11 was truncated spherical cap. Its height is 6. What part of the volume is a spherical cap from the whole sphere?
• Spherical cap
  Place a part of the sphere on a 4.6 cm cylinder so that the surface of this section is 20 cm². Determine the radius r of the sphere from which the spherical cap was cut.
• Elevation
  What must be the elevation of an observer in order that he may be able to see an object on the earth 536 km away? Assume the earth to be a smooth sphere with radius 6378.1 km.
• Spherical cap
  The spherical cap has a base radius of 8 cm and a height of 5 cm. Calculate the radius of a sphere of which this spherical cap is cut.
• Airplane
  Aviator sees part of the earth's surface with an area of 200,000 square kilometers. How high he flies?
• Spherical cap
  What is the surface area of a spherical cap, the base diameter 20 m, height 2 m.
• Sphere - parts
  Calculate the area of a spherical cap, which is part of an area with base radius ρ = 9 cm and a height v = 3.1 cm.
• 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?
• 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.
• The observatory
  The dome of the hemisphere-shaped observatory is 5.4 meters high. How many square meters of sheet metal needs to be covered to cover it, and 15 percent must be added to the minimum amount due to joints and waste?
• Spherical segment
  The spherical segment with height h=1 has a volume V=223. Calculate the radius of the sphere of which is cut this segment.
• What percentage
  What percentage of the Earth’s surface is seen by an astronaut from a height of h = 350 km. Take the Earth as a sphere with the radius R = 6370 km
• Spherical section cut
  Find the volume of a spherical section if the radius of its base is 10 cm and the magnitude of the central angle ω = 120 degrees.
• Cube and sphere
  Cube with the surface area 150 cm² is described sphere. What is sphere surface?
• Felix
  Calculate how much land saw Felix Baumgartner after jump from 32 km above ground. The radius of the Earth is R = 6378 km.