The radius and height of a right circular cone are in the ratio of 5:12 . If its volume is `314""(2)/(7) m^3`, its Slant height is :

Options

A. 26 m
B. 19.5 m
C. 13 m
D. none of them

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the given ratio
The radius (r) and height (h) of the cone are in the ratio of 5:12. We can express the radius and height in terms of a variable \( x \):
- Radius \( r = 5x \)
- Height \( h = 12x \)

### Step 2: Write down the formula for the volume of a cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]

### Step 3: Substitute the values into the volume formula
We know the volume of the cone is \( 314 \frac{2}{7} \) m³. First, convert this mixed number into an improper fraction:
\[ 314 \frac{2}{7} = \frac{314 \times 7 + 2}{7} = \frac{2198 + 2}{7} = \frac{2200}{7} \]
Now substitute \( r = 5x \) and \( h = 12x \) into the volume formula:
\[ \frac{2200}{7} = \frac{1}{3} \pi (5x)^2 (12x) \]

### Step 4: Simplify the equation
Calculating \( (5x)^2 \):
\[ (5x)^2 = 25x^2 \]
Now substitute back into the volume equation:
\[ \frac{2200}{7} = \frac{1}{3} \pi (25x^2)(12x) \]
This simplifies to:
\[ \frac{2200}{7} = \frac{1}{3} \pi (300x^3) \]
\[ \frac{2200}{7} = 100\pi x^3 \]

### Step 5: Solve for \( x^3 \)
To isolate \( x^3 \), multiply both sides by 3:
\[ \frac{6600}{7} = 300\pi x^3 \]
Now divide both sides by \( 300\pi \):
\[ x^3 = \frac{6600}{7 \times 300\pi} \]
Calculating \( 7 \times 300 = 2100 \):
\[ x^3 = \frac{6600}{2100\pi} \]
This simplifies to:
\[ x^3 = \frac{22}{7\pi} \]

### Step 6: Find the value of \( x \)
To find \( x \), we take the cube root:
\[ x = \sqrt[3]{\frac{22}{7\pi}} \]
For practical purposes, we can use the approximate value of \( \pi \approx 3.14 \):
\[ x \approx \sqrt[3]{\frac{22}{21.98}} \approx 1 \text{ m} \]

### Step 7: Calculate the radius and height
Now substituting \( x = 1 \) back into the expressions for radius and height:
- Radius \( r = 5x = 5 \times 1 = 5 \) m
- Height \( h = 12x = 12 \times 1 = 12 \) m

### Step 8: Calculate the slant height
The slant height \( l \) of a cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Substituting the values:
\[ l = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ m} \]

### Final Answer
The slant height of the cone is \( 13 \) meters.

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