Geometry

Solids

Medium

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?

2 percent decrease

2.5 percent decrease

2.5 percent increase

2 percent increase

✅ Correct Option: 4

The curved surface area (CSA) of a cone is given by:

$\text{CSA} = \pi r l$

where $r$ is the radius of the base and $l$ is the slant height.

Let the original radius be $r$ and the original slant height be $l$ .

Original CSA $= \pi r l$

The slant height decreases by 15%, so the new slant height is 85% of the original:

New slant height $= 0.85l$

The radius increases by 20%, so the new radius is 120% of the original:

New radius $= 1.20r$

The new curved surface area is:

New CSA $= \pi \times (1.20r) \times (0.85l)$

New CSA $= \pi \times 1.20 \times 0.85 \times r \times l$

New CSA $= 1.02 \times \pi r l$

The new CSA is 1.02 times the original CSA, which represents 102% of the original.

Percentage change $= (1.02 - 1) \times 100\%$

Percentage change $= 0.02 \times 100\%$

Percentage change $= 2\%$

Since the new CSA is greater than the original, this represents a 2 percent increase.

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