Question

The area of a circle varies directly as the square of its radius. A circle with radius 2 in. has area 12.568 in. squared. What is the area of a circle with radius 6.2 ​in.?
Question content area bottom
Part 1
The area of a circle with radius 6.2 in. is approximately
  
120.634
in. squared .
​(Simplify your answer. Round to the nearest thousandth as​ needed.)

Answer 1

The area \( A \) of a circle varies directly as the square of its radius \( r \). This can be expressed with the formula:

\[
A = k r^2
\]

where \( k \) is the constant of proportionality. Given that the area of a circle with radius 2 inches is 12.568 square inches, we can first find \( k \):

\[
12.568 = k (2^2)
\]

\[
12.568 = k (4)
\]

\[
k = \frac{12.568}{4} = 3.142
\]

Now that we have the value of \( k \), we can use it to find the area of a circle with a radius of 6.2 inches:

\[
A = k r^2
\]
\[
A = 3.142 (6.2^2)
\]

Calculating \( 6.2^2 \):

\[
6.2^2 = 38.44
\]

Now, substituting back into the area formula:

\[
A = 3.142 \times 38.44
\]

Calculating this gives:

\[
A \approx 120.63408
\]

Rounding this to the nearest thousandth, the area of a circle with a radius of 6.2 inches is approximately:

\[
A \approx 120.634 \text{ in}^2
\]