Question

The semi-vertical angle of a right circular cone is \( 45^\circ \). If the radius of the base of the cone is measured as 14 cm with an error of \( \left(\frac{\sqrt{2}-1}{11} \right) \) cm, then the approximate error in measuring its total surface area is (in sq. cm).

• A. \( 14 \)
• B. \( 8 \)
• C. \( 5 \)
• D. \( 4 \)

Hint:

For approximating errors in surface area, use differentiation and multiply the derivative by the given error in measurement.

Answer 1

The Correct Option is B

Step 1: Formula for total surface area of a cone
The total surface area of a right circular cone is given by: \[ A = \pi r (r + l), \] where \( r \) is the radius and \( l \) is the slant height. Since the semi-vertical angle is \( 45^\circ \), we have: \[ \tan 45^\circ = \frac{r}{h} = 1 \Rightarrow r = h. \] Using the Pythagorean theorem: \[ l = \sqrt{r^2 + h^2} = \sqrt{r^2 + r^2} = \sqrt{2} r. \] Thus, the total surface area is: \[ A = \pi r (r + \sqrt{2} r) = \pi r^2 (1 + \sqrt{2}). \] Step 2: Approximate error in surface area
The error in total surface area is given by: \[ dA = \frac{dA}{dr} \cdot dr. \] Differentiating \( A \) with respect to \( r \): \[ \frac{dA}{dr} = 2\pi r (1 + \sqrt{2}). \] Substituting \( r = 14 \) cm: \[ \frac{dA}{dr} = 2\pi (14) (1 + \sqrt{2}). \] The given error in radius is: \[ dr = \frac{\sqrt{2} - 1}{11}. \] Thus, the approximate error in area is: \[ dA = 2\pi (14) (1 + \sqrt{2}) \times \frac{\sqrt{2} - 1}{11}. \] Evaluating, we get: \[ dA \approx 8 \text{ sq. cm}. \] Step 3: Conclusion
Thus, the final answer is: \[ \boxed{8}. \]