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Let's break down and solve the problem in the image.

**Problem 5:**
- **Find the volume \( V \) of water in the conical tank as a function of the height \( h \).**
- Then, express the height \( h \) as a function of the volume \( V \).

### Step 1: Understand the shape and dimensions
We are dealing with a conical tank with the following dimensions:
- Total height of the cone = 6 feet.
- Radius at the top of the cone = 4 feet.

The volume of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
where:
- \( r \) is the radius of the cone at a certain height \( h \).
- \( h \) is the height of the water.

### Step 2: Relating the radius to the height
Since the cone is similar at all heights, the radius \( r \) at any height \( h \) is proportional to \( h \). The ratio of the radius to the height remains constant throughout the cone.

Using the full dimensions of the cone, the radius-to-height ratio is:
\[
\frac{r}{h} = \frac{4}{6} = \frac{2}{3}
\]
Thus, for any height \( h \), the radius \( r \) can be expressed as:
\[
r = \frac{2}{3}h
\]

### Step 3: Substitute this into the volume formula
Substitute \( r = \frac{2}{3}h \) into the formula for the volume of a cone:
\[
V = \frac{1}{3} \pi \left( \frac{2}{3}h \right)^2 h
\]
Simplify the equation:
\[
V = \frac{1}{3} \pi \frac{4}{9} h^3 = \frac{4\pi}{27} h^3
\]

### Step 4: Express height \( h \) as a function of volume \( V \)
Now, to express \( h \) as a function of \( V \), solve for \( h \) in the volume equation:
\[
V = \frac{4\pi}{27} h^3
\]
Multiply both sides by \( \frac{27}{4\pi} \):
\[
h^3 = \frac{27V}{4\pi}
\]
Take the cube root of both sides:
\[
h = \left( \frac{27V}{4\pi} \right)^{\frac{1}{3}}
\]

### Final Answer:
1. Volume \( V \) as a function of height \( h \):
   \[
   V = \frac{4\pi}{27} h^3
   \]

2. Height \( h \) as a function of volume \( V \):
   \[
   h = \left( \frac{27V}{4\pi} \right)^{\frac{1}{3}}
   \]

Would you like further details on any part of the solution?

Here are 5 related questions you might find interesting:
1. How would the volume formula change if the cone was not symmetrical?
2. What happens to the volume formula if the cone is filled halfway?
3. Can you derive a similar formula for the volume of water in a cylindrical tank?
4. How would the result change if the cone had a different proportionality ratio between radius and height?
5. What if the cone had an elliptical base instead of a circular one?

**Tip:** Always look for proportional relationships in geometric figures. These relationships simplify calculations and make formulas easier to derive.

Find the volume V of water in the conical tank as a function of the height h. Then express the height h as a function of the volume V.

Learn how to find the volume of water in a conical tank as a function of its height and express the height in terms of the volume. This involves geometric principles, proportional relationships, and the formula for the volume of a cone. Suitable for Grades 11-12.

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