Circumference of a Circle The circumference of a circle varies directly as the radius. A circle with radius 7 in. has circumference 43.96 in. Find the circumference of the circle if the radius changes to 11 in.

Verified step by step guidance

1. Identify the direct variation relationship between circumference (C) and radius (r). Since circumference varies directly as radius, we can write the equation as \(C = k \times r\), where \(k\) is the constant of proportionality.
2. Use the given values to find the constant \(k\). Substitute \(C = 43.96\) and \(r = 7\) into the equation: \(43.96 = k \times 7\).
3. Solve for \(k\) by dividing both sides of the equation by 7: \(k = \frac{43.96}{7}\).
4. Write the general formula for circumference using the found constant: \(C = k \times r\).
5. Find the new circumference when the radius is 11 by substituting \(r = 11\) into the formula: \(C = k \times 11\).

Key Concepts

Direct Variation

Direct variation describes a relationship where one quantity changes proportionally with another. If y varies directly as x, then y = kx for some constant k. In this problem, the circumference varies directly as the radius, meaning circumference = k × radius.

Finding the Constant of Variation

To solve direct variation problems, first find the constant k by using known values. Here, use the given radius and circumference to calculate k = circumference ÷ radius. This constant allows you to find the circumference for any other radius.

Applying the Variation to Find Unknown Values

Once the constant k is known, substitute the new radius into the direct variation formula to find the new circumference. This step involves simple multiplication and helps solve for the unknown quantity based on the established relationship.