The ratio of the heights of two cones of same base is 4 : 3. If the volume of small cone is `462 pi cm^(3)`, then find the volume of new cone.

Answer

To solve the problem step by step, we will use the properties of cones and the given ratios.

### Step 1: Understand the given information
We know that the ratio of the heights of two cones is 4:3. We also know the volume of the smaller cone is \(462 \pi \, \text{cm}^3\).

### Step 2: Define the heights of the cones
Let the height of the new cone be \(h_1\) and the height of the smaller cone be \(h_2\). According to the ratio given:
\[ h_1 : h_2 = 4 : 3 \]
This means we can express the heights as:
\[ h_1 = 4k \quad \text{and} \quad h_2 = 3k \]
for some constant \(k\).

### Step 3: Use the formula for the volume of a cone
The volume \(V\) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Since both cones have the same base, their radii \(r\) are equal.

### Step 4: Set up the volume ratio
The volumes of the two cones can be expressed as:
- Volume of the new cone:
\[ V_1 = \frac{1}{3} \pi r^2 h_1 = \frac{1}{3} \pi r^2 (4k) \]
- Volume of the smaller cone:
\[ V_2 = \frac{1}{3} \pi r^2 h_2 = \frac{1}{3} \pi r^2 (3k) \]

### Step 5: Find the ratio of the volumes
The ratio of the volumes \(V_1\) to \(V_2\) is:
\[ \frac{V_1}{V_2} = \frac{\frac{1}{3} \pi r^2 (4k)}{\frac{1}{3} \pi r^2 (3k)} = \frac{4k}{3k} = \frac{4}{3} \]

### Step 6: Relate the volumes to the known volume
We know the volume of the smaller cone \(V_2 = 462 \pi \, \text{cm}^3\). Therefore, we can write:
\[ \frac{V_1}{462 \pi} = \frac{4}{3} \]

### Step 7: Solve for the volume of the new cone
Cross-multiplying gives:
\[ V_1 = \frac{4}{3} \times 462 \pi \]
Calculating \(V_1\):
\[ V_1 = \frac{4 \times 462}{3} \pi \]
Calculating \(4 \times 462\):
\[ 4 \times 462 = 1848 \]
Now divide by 3:
\[ \frac{1848}{3} = 616 \]
Thus, the volume of the new cone is:
\[ V_1 = 616 \pi \, \text{cm}^3 \]

### Final Answer
The volume of the new cone is \(616 \pi \, \text{cm}^3\).

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