https://file.aiquickdraw.com/mathbot/file/1725513118443yy0kxgrn.jpg

The problem in the image involves the volume \( V \) of a sphere in terms of its radius \( r \). The volume formula is given by:

\[
V(r) = \frac{4}{3} \pi r^3
\]

The task is to express the radius \( r \) as a function of the volume \( V \) and then to find the radius of a sphere with a given volume of 400 cubic feet.

### Step 1: Solving for \( r \)
To express \( r \) in terms of \( V \), we start with the formula for the volume of a sphere:

\[
V = \frac{4}{3} \pi r^3
\]

Rearranging this equation to solve for \( r \):

\[
r^3 = \frac{3V}{4\pi}
\]

Now take the cube root of both sides:

\[
r = \sqrt[3]{\frac{3V}{4\pi}}
\]

### Step 2: Plugging in the given volume
The volume is given as \( V = 400 \, \text{cubic feet} \). Plug this value into the equation for \( r \):

\[
r = \sqrt[3]{\frac{3(400)}{4\pi}}
\]

Now let's calculate the radius numerically. I will compute this for you.The radius of the sphere with a volume of 400 cubic feet is approximately 4.57 feet.

Would you like further clarification or details on any part of the process?

Here are five related questions to deepen your understanding:
1. How would the radius change if the volume of the sphere doubled?
2. Can you derive the formula for the surface area of a sphere from the volume formula?
3. What is the relationship between the surface area and volume of a sphere?
4. How would you express the diameter of the sphere in terms of its volume?
5. What is the volume of a sphere with a radius of 10 feet?

**Tip:** When dealing with spherical objects, remember that volume grows much faster than surface area as the radius increases.

The volume V of a sphere in terms of its radius r is given by V(r) = (4/3)πr^3. Express r as a function of V and find the radius of a sphere with volume 400 cubic feet.

Solve for the radius of a sphere given the volume using the formula V = (4/3)πr^3. This problem involves fundamental geometry and algebra concepts, suitable for high school students (Grades 9-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation