Bunuel wrote:

If the radius of a circle is decreased by 30 percent, by what percent will the area of the circular region be decreased?

(A) 15%

(B) 49%

(C) 51%

(D) 60%

(E) 90%

Choose valuesLet the original radius = 10

Original area is \(\pi*r^2=100\pi\)

The radius is decreased by 30 percent:

.70(10) = 7

New area is \(\pi*r^2=49\pi\)

Percent decrease:

\(\frac{New-Old}{Old} * 100\)

\(\frac{100-49}{100}=\frac{51}{100} * 100 = 51\) percent

Answer C

Scale FactorLength * Length = Area

Scale factor = (1 - .30) = .70

When area (length * length) is decreased by a scale factor, the scale factor, squared (because it is used for both lengths)*, can be used to calculate the percent change.

\((.7)^2 = .49\)

Original = \(1\)

Percent decrease:

\((1 - .49) = .51 * 100 = 51\) percent

Answer C

That is:

Length * length = area = (scale factor)\(^2\) for change

\(\pi\) is constant

Radius is the length affected twice by scale factor.

Original, \(r^2=(r)(r)\) to new, \((.7r)^2=(.7r)(.7r)\)
_________________

In the depths of winter, I finally learned

that within me there lay an invincible summer.

-- Albert Camus, "Return to Tipasa"