Problem 9 - Calculus

Question

Find the distance \(d\) from (1,0) to \(\left(\frac{1}{2}, \sqrt{3} / 2\right)\) and show on a circle why \(6 d\) is less than \(2 \pi\).

Step-by-Step Solution

Answer

The distance is 1; \(6d = 6\), which is less than \(2\pi\).

Step 1: Write the Distance Formula

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Step 2: Substitute the Coordinates

Substitute the given coordinates \((x_1, y_1) = (1, 0)\) and \((x_2, y_2) = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) into the distance formula:\[d = \sqrt{\left(\frac{1}{2} - 1\right)^2 + \left(\frac{\sqrt{3}}{2} - 0\right)^2}\]

Step 3: Simplify the Expressions

Calculate the differences inside the square roots:\(\frac{1}{2} - 1 = -\frac{1}{2}\) \(\frac{\sqrt{3}}{2} - 0 = \frac{\sqrt{3}}{2}\)

Step 4: Calculate the Distance

Substitute these differences into the distance formula:\[d = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2}\]Simplify further:\[d = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1\]

Step 5: Find 6 Times the Distance

Now, calculate \(6d\):\[6d = 6 \times 1 = 6\]

Step 6: Compare to Circle Circumference

The circumference of the unit circle, where the radius \(r = 1\), is given by the formula:\[C = 2\pi r = 2\pi \times 1 = 2\pi\]Comparing \(6d\) to this circumference, we have:\[6d = 6 < 2\pi \approx 6.28\]

Key Concepts

Distance between points

To find the distance between two points on a coordinate plane, we use the distance formula. This fundamental concept in geometry helps us calculate the straight-line distance between any pair of points given their coordinates. The distance formula is derived from the Pythagorean Theorem, which relates the three sides of a right triangle. When you have two points, say

• \((x_1, y_1)\) and
• \((x_2, y_2)\),

you can find their distance \(d\) using:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula is essentially calculating the hypotenuse of a right triangle formed by the differences in the x-coordinates and y-coordinates of the points, making it a practical tool for measuring distance in the plane.

Circle circumference

The concept of circle circumference is all about the total distance around a circle. Depending on the circle's radius, the circumference allows us to relate the concept of circular distance with linear measurements. For a circle with a radius \(r\), the circumference \(C\) is calculated using the formula: \[C = 2\pi r\]In the specific exercise, we consider the unit circle where the radius \(r = 1\). Thus, the circumference simplifies to: \[C = 2\pi \] This concept becomes especially relevant when we need to compare the linear distance found using the distance formula to a curved distance, such as the perimeter of a circle. Understanding this relationship aids in visualizing how regular distances compare within circular or curvilinear spaces.

Coordinate geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to discuss and solve geometric problems using the coordinate plane. It involves plotting points on a plane using ordered pairs

• \((x, y)\)

and analyzing their spatial relationships.Coordination between algebraic equations and geometric figures enables us to represent shapes like lines and circles mathematically, facilitating precise calculations for slopes, distances, and intersections. In the given problem, coordinate geometry allows us to evaluate the spatial relationship between two distinct points in the plane and establish a comparison between a line segment’s length and a circle's circumference. This field of mathematics is powerful as it gives us tools to logically quantify and analyze spatial data in a visually approachable manner. Understanding coordinate geometry is vital for solving complex geometry problems that involve diverse figures and their interrelationships.