Question

The total surface area of a cone, whose radius is \(\frac{r}{2}\) and slant height is \(2l\), will be :

• A. \(2\pi r(l+r)\)
• B. \(\pi r(l + \frac{r}{4})\)
• C. \(\pi r(l+r)\)
• D. \(2\pi rl\)

Hint:

1. Write down the general formula for the total surface area of a cone: TSA = \(\pi \times (\text{actual radius}) \times (\text{actual radius} + \text{actual slant height})\). 2. Identify the given actual radius and actual slant height from the problem: Actual radius = \(r/2\) Actual slant height = \(2l\) 3. Substitute these into the formula: TSA = \(\pi \left(\frac{r}{2}\right) \left(\frac{r}{2} + 2l\right)\). 4. Simplify the expression: \(\pi \frac{r}{2} (2l + \frac{r}{2}) = \pi rl + \frac{\pi r^2}{4} = \pi r(l + \frac{r}{4})\).

Answer 1

The Correct Option is B

Concept: The total surface area (TSA) of a cone is the sum of the area of its circular base and its curved surface area (CSA). The formulas are:
Area of base = \(\pi \times (\text{radius})^2\)
Curved Surface Area (CSA) = \(\pi \times \text{radius} \times \text{slant height}\)
Total Surface Area (TSA) = \(\pi \times \text{radius}^2 + \pi \times \text{radius} \times \text{slant height}\) TSA = \(\pi \times \text{radius} \times (\text{radius} + \text{slant height})\) Step 1: Identify the given dimensions of the cone Let the actual radius of the cone be \(R_{cone}\) and its actual slant height be \(L_{cone}\). Given:
Radius of the cone, \(R_{cone} = \frac{r}{2}\)
Slant height of the cone, \(L_{cone} = 2l\) Here, \(r\) and \(l\) are general variables used in the problem statement and options. Step 2: Substitute these dimensions into the TSA formula The general formula for TSA of a cone is TSA = \(\pi R_{cone} (R_{cone} + L_{cone})\). Substitute \(R_{cone} = \frac{r}{2}\) and \(L_{cone} = 2l\): \[ \text{TSA} = \pi \left(\frac{r}{2}\right) \left( \frac{r}{2} + 2l \right) \] Step 3: Simplify the expression First, rearrange the terms inside the second parenthesis to match common order: \[ \text{TSA} = \pi \frac{r}{2} \left( 2l + \frac{r}{2} \right) \] Now, distribute the \(\pi \frac{r}{2}\) term: \[ \text{TSA} = \left(\pi \frac{r}{2} \times 2l\right) + \left(\pi \frac{r}{2} \times \frac{r}{2}\right) \] \[ \text{TSA} = \pi r l + \pi \frac{r^2}{4} \] Factor out \(\pi r\) from both terms: \[ \text{TSA} = \pi r \left( l + \frac{r}{4} \right) \] Step 4: Compare with the given options The simplified expression is \(\pi r(l + \frac{r}{4})\). This matches option (2).