Question

If the diameter of a circle is increased by \(10 \%\), by how much does the area of the circle increase? A. \(5 \%\) B. \(10.5 \%\) C. \(14 \%\) D. \(17.5 \%\) E. \(21 \%\)

Short answer

The area of the circle increases by 21%. Answer: E.

Step-by-step solution

Understand the relationship between diameter and area of a circle

The area of a circle is given by the formula \(\text{Area} = \frac{\text{π} (\text{diameter})^2}{4}\). If the diameter is denoted by \(d\), then \(\text{Area} = \frac{\text{π} d^2}{4}\).

Express the new diameter

If the diameter is increased by 10%, the new diameter becomes \(d_{new} = 1.1d\).

Calculate the new area

Substitute \(d_{new}\) into the area formula: \(\text{Area}_{new} = \frac{\text{π} (1.1d)^2}{4}\). Simplify this expression to get the new area: \(\text{Area}_{new} = \frac{\text{π} \times 1.21d^2}{4} = 1.21 \times \frac{\text{π} d^2}{4}\).

Find the percentage increase in the area

The original area is \( \text{Area}_{original} = \frac{\text{π} d^2}{4}\). The increase in area is \( \text{Area}_{new} - \text{Area}_{original} = 1.21 \times \text{Area}_{original} - \text{Area}_{original} \). This simplifies to \(0.21 \times \text{Area}_{original}\), which indicates a 21% increase in the area.

Key concepts

Circle Geometry

Circle geometry involves understanding the properties and measurements of circles. A circle is a shape with all points at a constant distance from its center. This distance is called the radius (r). One important property is the diameter (d), which is twice the radius: \(d = 2r\). Another key measurement is the circumference, which represents the circle's perimeter, and can be calculated using \(2 \pi r\). The area of a circle can be determined with the formula \(\pi r^2 \). These formulas are fundamental for solving problems related to circle geometry.

Percentage Increase

Calculating percentage increase is crucial when comparing how a quantity grows. To calculate the percentage increase, subtract the original value from the new value, divide by the original value, and then multiply by 100 to get the percentage.

For example, if we have a diameter that is increased by 10%, we start by determining the new diameter: \( \text{new diameter} = \text{original diameter} \times 1.1 \). This increment is essential for understanding changes in both linear dimensions and more complex attributes like area or volume.

Area Calculation

Area calculation is crucial for understanding how much space a shape covers. For circles, the area is given by \(\pi r^2\). When dimensions change, the area calculation must be adjusted accordingly.

For instance, if the diameter of a circle increases by 10%, we must first find the new radius. If the original radius was r, the new radius is \(1.1r\). Then, the new area is \(\pi (1.1r)^2 = 1.21\pi r^2\). Comparing this to the original area, which is \(\pi r^2\), we see the area increases by 21%.

Mathematical Relationships

Understanding mathematical relationships helps solve complex problems efficiently. Variables in geometry are often interrelated, and changing one affects others predictably.

In our case, the area of a circle depends on its radius or diameter. Increasing the diameter by 10% increases the area by a compounded factor of that change. This relationship is quadratic because the diameter squares in the area formula. Recognizing this helps anticipate how changes in one dimension impact another, essential for solving problems involving proportional changes.