Asked by AI
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 5.7 in.?
Answers
Answered by
GPT-4o mini
The area \( A \) of a circle is given by the formula:
\[
A = k r^2
\]
where \( k \) is the constant of proportionality, and \( r \) is the radius of the circle.
From the information provided, we know that for a circle with a radius of 2 inches, the area is 12.568 square inches. We can use this information to find \( k \).
Substituting the values we have:
\[
12.568 = k \cdot (2^2)
\]
\[
12.568 = k \cdot 4
\]
Solving for \( k \):
\[
k = \frac{12.568}{4} = 3.142
\]
Now that we have the value of \( k \), we can find the area of a circle with a radius of 5.7 inches.
Using the formula again:
\[
A = k r^2
\]
Substituting \( k = 3.142 \) and \( r = 5.7 \):
\[
A = 3.142 \cdot (5.7^2)
\]
Calculating \( 5.7^2 \):
\[
5.7^2 = 32.49
\]
Then substituting back into the area formula:
\[
A = 3.142 \cdot 32.49 \approx 102.12658
\]
Thus, the area of a circle with a radius of 5.7 inches is approximately \( 102.13 \) square inches.
\[
A = k r^2
\]
where \( k \) is the constant of proportionality, and \( r \) is the radius of the circle.
From the information provided, we know that for a circle with a radius of 2 inches, the area is 12.568 square inches. We can use this information to find \( k \).
Substituting the values we have:
\[
12.568 = k \cdot (2^2)
\]
\[
12.568 = k \cdot 4
\]
Solving for \( k \):
\[
k = \frac{12.568}{4} = 3.142
\]
Now that we have the value of \( k \), we can find the area of a circle with a radius of 5.7 inches.
Using the formula again:
\[
A = k r^2
\]
Substituting \( k = 3.142 \) and \( r = 5.7 \):
\[
A = 3.142 \cdot (5.7^2)
\]
Calculating \( 5.7^2 \):
\[
5.7^2 = 32.49
\]
Then substituting back into the area formula:
\[
A = 3.142 \cdot 32.49 \approx 102.12658
\]
Thus, the area of a circle with a radius of 5.7 inches is approximately \( 102.13 \) square inches.