Sphere radius

The radius of the sphere we reduce by 1/3 of the original radius. How much percent does the volume and surface of the sphere change?

Correct answer:

p₁ =  70.3704 %
p₂ =  55.5556 %

Step-by-step explanation:

$$\begin{gathered} r_1=1 \\ {r}_2=r_1-r_1\mathrm{/}3=1-1\mathrm{/}3={\displaystyle \frac{2}{3}}\doteq 0.6667 \\ {V}_1={\displaystyle \frac{4}{3}}\cdot \pi \cdot r_1^3={\displaystyle \frac{4}{3}}\cdot 3.1416\cdot 1^3\doteq 4.1888 \\ {V}_2={\displaystyle \frac{4}{3}}\cdot \pi \cdot r_2^3={\displaystyle \frac{4}{3}}\cdot 3.1416\cdot 0.6667^3\doteq 1.2411 \\ {p}_1=100\cdot {\displaystyle \frac{V_1-V_2}{V_1}}=100\cdot {\displaystyle \frac{4.1888-1.2411}{4.1888}}={\displaystyle \frac{1900}{27}}\mathrm{\%}=70.3704\mathrm{\%} \end{gathered}$$

$$\begin{gathered} S_1=4\pi \cdot r_1^2=4\cdot 3.1416\cdot 1^2\doteq 12.5664 \\ {S}_2=4\pi \cdot r_2^2=4\cdot 3.1416\cdot 0.6667^2\doteq 5.5851 \\ {p}_2=100\cdot {\displaystyle \frac{S_1-S_2}{S_1}}=100\cdot {\displaystyle \frac{12.5664-5.5851}{12.5664}}={\displaystyle \frac{500}{9}}\mathrm{\%}=55.5556\mathrm{\%} \end{gathered}$$

Try another example

Related math problems and questions:

• Twice of radius
  How many times does the surface of a sphere decrease if we reduce its radius twice?
• The ball
  The ball has a radius of 2m. What percentage of the surface and volume is another sphere whose radius is 20% larger?
• Sphere vs cube
  How many % of the surface of a sphere of radius 12 cm is the surface of a cube inscribed in this sphere?
• Sphere
  Intersect between plane and a sphere is a circle with a radius of 60 mm. Cone whose base is this circle and whose apex is at the center of the sphere has a height of 34 mm. Calculate the surface area and volume of a sphere.
• Inscribed sphere
  How many % of the volume of the cube whose edge is 6 meters long is a volume of a sphere inscribed in that cube?
• Surface area 2
  Calculate how many % reduce the surface area of the cube is we reduced length of each edge by 10%.
• 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?
• Cube in sphere
  The cube is inscribed in a sphere with a radius r = 6 cm. What percentage is the volume of the cube from the volume of the ball?
• Reduce of the volume
  Calculate how many % reduce the volume of the cube is we reduced length of each edge by 10%.
• 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.
• Cube, cuboid, and sphere
  Volumes of a cube and a cuboid are in ratio 3: 2. Volumes of sphere and cuboid are in ratio 1: 3. At what rate are the volumes of cube, cuboid, and sphere?
• Cube zoom
  How many percents do we increase the volume and surface of the cube if we magnify its edge by 38 %?
• Sphere cut
  A sphere segment is cut off from a sphere k with radius r = 1. The volume of the sphere inscribed in this segment is equal to 1/6 of the segment's volume. What is the distance of the cutting plane from the center of the sphere?
• Percent change
  If the length of a rectangle is increased by 25% and the width is decreased by 10%, the area of the rectangle is larger than the area of the original rectangle by what percent?
• Equilateral cylinder
  A sphere is inserted into the rotating equilateral cylinder (touching the bases and the shell). Prove that the cylinder has both a volume and a surface half larger than an inscribed sphere.
• 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.
• Earth's surface
  The greater part of the earth's surface (r = 6371 km) is covered by oceans; their area is approximately 71% of the Earth's surface. What is the approximate area of the land?