`V =1/3 pi ((d)/(2)) ^(2) h`
A circle of rubber with a constant diameter d is placed on a table, its perimeter is anchored to the table and a string is attached to its centre. When the string is pulled upwards, a cone is formed with height h and volume V. The relationship between d, h, and V is represented above. Which of the following statements must be true ?
I. As the volume of the cone decreases, the height also decreases.
II. If the diameter of the base of the cone is 6 centimeters, the height can be determined by dividing the volume by `3pi.`
III. If the height of the conc triples, the volume must also triple.

Options

A. I only
B. I and II only
C. II and III only
D. I, II, and III

Explanation

To solve the problem, we need to analyze the given equation for the volume of the cone and evaluate the truth of each statement based on that equation.

### Given Equation:
The volume \( V \) of the cone is given by:
\[ V = \frac{1}{3} \pi \left(\frac{d}{2}\right)^2 h \]
where \( d \) is the diameter of the base of the cone, and \( h \) is the height of the cone.

### Step 1: Analyze Statement I
**Statement I:** As the volume of the cone decreases, the height also decreases.

Since the diameter \( d \) is constant, we can rewrite the equation as:
\[ V = k \cdot h \quad \text{where } k = \frac{1}{3} \pi \left(\frac{d}{2}\right)^2 \]
This shows that \( V \) is directly proportional to \( h \). Therefore, if \( V \) decreases, \( h \) must also decrease.

**Conclusion for Statement I:** True.

### Step 2: Analyze Statement II
**Statement II:** If the diameter of the base of the cone is 6 centimeters, the height can be determined by dividing the volume by \( 3\pi \).

Let’s substitute \( d = 6 \) cm into the volume equation:
\[ V = \frac{1}{3} \pi \left(\frac{6}{2}\right)^2 h = \frac{1}{3} \pi \cdot 3^2 \cdot h = \frac{1}{3} \pi \cdot 9 \cdot h = 3\pi h \]
To find \( h \), we rearrange the equation:
\[ h = \frac{V}{3\pi} \]

**Conclusion for Statement II:** True.

### Step 3: Analyze Statement III
**Statement III:** If the height of the cone triples, the volume must also triple.

From the volume equation:
\[ V = k \cdot h \]
If \( h \) triples (i.e., \( h \) becomes \( 3h \)), then:
\[ V' = k \cdot (3h) = 3(k \cdot h) = 3V \]
This shows that if the height triples, the volume also triples.

**Conclusion for Statement III:** True.

### Final Conclusion
Since all three statements are true, the correct answer is:
**Option D:** First, second, and third are all true.