The radius of base and slant height of a cone are in the ratio 4 : 7. If its curved surface area is 792 `cm^(2)`, then the radius (in cm) of its base is [Use `pi = 22//7`]

Options

A. 8
B. 12
C. 14
D. 16

Answer

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

### Step 1: Understand the given ratio
The radius (r) and the slant height (l) of the cone are in the ratio of 4:7. We can express this as:
- r = 4x
- l = 7x
where x is a common multiplier.

**Hint:** Remember that the ratio gives us a way to express two quantities in terms of a single variable.

### Step 2: Write the formula for the curved surface area of a cone
The formula for the curved surface area (CSA) of a cone is given by:
\[ \text{CSA} = \pi r l \]

**Hint:** Make sure you know the formula for the curved surface area of a cone, as it is essential for solving this problem.

### Step 3: Substitute the values into the formula
Substituting the expressions for r and l into the CSA formula, we get:
\[ \text{CSA} = \pi (4x)(7x) \]
\[ \text{CSA} = \pi \cdot 28x^2 \]

**Hint:** When substituting, ensure that you multiply the expressions correctly.

### Step 4: Set the equation equal to the given CSA
We know that the curved surface area is 792 cm². Therefore, we can set up the equation:
\[ 28\pi x^2 = 792 \]

**Hint:** Always equate the expression you derived to the given value in the problem.

### Step 5: Substitute the value of π
Using the value of π as \( \frac{22}{7} \):
\[ 28 \cdot \frac{22}{7} \cdot x^2 = 792 \]

**Hint:** Be careful with the fractions when substituting values.

### Step 6: Simplify the equation
Multiply both sides by 7 to eliminate the fraction:
\[ 28 \cdot 22 \cdot x^2 = 792 \cdot 7 \]
\[ 616x^2 = 5544 \]

**Hint:** Multiplying both sides by the same number keeps the equation balanced.

### Step 7: Solve for x²
Now, divide both sides by 616:
\[ x^2 = \frac{5544}{616} \]
\[ x^2 = 9 \]

**Hint:** When dividing, ensure that you simplify the fraction correctly.

### Step 8: Find the value of x
Taking the square root of both sides:
\[ x = 3 \]

**Hint:** Remember that we are looking for the positive value since x represents a length.

### Step 9: Calculate the radius
Now, substitute the value of x back to find the radius:
\[ r = 4x = 4 \cdot 3 = 12 \, \text{cm} \]

**Hint:** Always substitute back to find the required quantity after finding the variable.

### Final Answer
The radius of the base of the cone is **12 cm**.