Question

The area of a circle is directly
proportional to the square of its
radius. A circle with a radius of 2
cm has an area of 12.566 cm^2.
What is the radius of a circle with
an area of 78.54 cm^2?
[?] cm
Round to the nearest whole number.

Answer 1

Answer:

$radius=5\ cm$

Step-by-step explanation:

Let r be the radius of the circle and A be the area of the circle.

Given:

$r_{1} = 2\ cm, A_{1}=12.566\ cm^{2}$

And $r_{2} = ?, A_{2}=78.54\ cm^{2}$

The area of a circle is directly  proportional to the square of its  radius.

A ∝ $r^{2}$

$A = kr^{2}$-----------(1)

Where k is the constant of proportionality

Find constant value by substituting $r_{1} = 2\ cm, A_{1}=12.566\ cm^{2}$

in equation 1.

$A = kr^{2}$

$A_{1}=k(r_{1})^{2}$

$12.566=k\times 2^{2}$

$12.566=k\times 4$

$k=\frac{12.566}{4}$

$k=3.141$

Find $r_{2}$ by substituting k and $A_{2}$ value in equation 1.

$A = kr^{2}$

$A_{2}=k(r_{2})^{2}$

$12.566=3.141\times (r_{2})^{2}$

$(r_{2})^{2}=\frac{78.54}{3.141}$

$(r_{2})^{2}=25.004$

where: 25.004 ≅ 25

$(r_{2})^{2}=25$

$r_{2}=\sqrt{25}$

$r_{2}=5\ cm$

Therefore; the radius of the circle is 5 cm.