If the slant height of a cone is decreased by 15 percent and the radius of its base is increased by 20 percent, then by what percent will its curved surface area change?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 2 percent increase
Step-by-Step Calculation:
1. Recall the Formula:
The Curved Surface Area (CSA) of a cone is calculated using the formula:
$CSA = \pi r l$
Where $r$ is the radius of the base and $l$ is the slant height.
2. Determine the Changes:
- Radius ($r$): Increased by 20%, so the new radius becomes $1.20r$.
- Slant Height ($l$): Decreased by 15%, so the new slant height becomes $0.85l$.
3. Calculate the New Curved Surface Area:
$\text{New CSA} = \pi \times (1.20r) \times (0.85l)$
$\text{New CSA} = (1.20 \times 0.85) \times \pi r l$
$\text{New CSA} = 1.02 \times \pi r l$
$\text{New CSA} = 1.02 \times \text{Original CSA}$
4. Find the Percentage Change:
- An area of $1.02$ times the original means an increase of $0.02$.
- $\text{Percentage Change} = 0.02 \times 100 = \mathbf{2\% \text{ increase}}$.