Math Problem Statement

the volume of a sphere is 36pie ft3. what is the radius is the sphere

Solution

The volume VV of a sphere is given by the formula:

V=43πr3V = \frac{4}{3} \pi r^3

Where:

  • VV is the volume of the sphere.
  • rr is the radius.
  • π\pi is Pi, approximately 3.14159.

You're given that the volume of the sphere is 36πft336\pi \, \text{ft}^3, so we can set up the equation:

36π=43πr336\pi = \frac{4}{3} \pi r^3

Now, divide both sides by π\pi to simplify:

36=43r336 = \frac{4}{3} r^3

Next, multiply both sides by 3 to eliminate the fraction:

108=4r3108 = 4r^3

Now, divide both sides by 4:

r3=27r^3 = 27

Finally, take the cube root of both sides to find rr:

r=273=3r = \sqrt[3]{27} = 3

So, the radius of the sphere is r=3ftr = 3 \, \text{ft}.

Would you like more details or have any questions? Here are some related questions to expand your understanding:

  1. What is the formula for the surface area of a sphere?
  2. How does the volume of a sphere change if its radius is doubled?
  3. What is the relationship between the volume of a sphere and a cube with the same radius?
  4. Can you derive the volume formula of a sphere using calculus?
  5. What is the difference in volume between two spheres with radii 3 ft and 6 ft?

Tip: When dealing with volume formulas, always make sure to work with consistent units for accurate calculations!

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Math Problem Analysis

Mathematical Concepts

Geometry
Volume
Spheres

Formulas

Volume of a sphere: V = (4/3)πr³

Theorems

Formula for the volume of a sphere

Suitable Grade Level

Grades 8-10