Solve each problem. Circumference of a CircleThe 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). This means 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 \(C = k \times r\) to get \$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. Substitute the new radius \(r = 11\) into the formula to find the new circumference: \(C = k \times 11\).

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, you first find the constant k by substituting known values into the equation y = kx. Here, using the given radius and circumference, you calculate k = circumference ÷ radius, which allows you to find the circumference for any other radius.

Applying the Direct Variation Formula

Once the constant k is known, you apply the formula circumference = k × radius to find the circumference for a new radius. This step involves substituting the new radius value into the equation and calculating the resulting circumference.