Problem 8 - Calculus

Question

Find the distance from (1,0) to (0,1) along (a) a straight line (b) a quarter- circle (c) a semicircle centered at \(\left(\frac{1}{2}, \frac{1}{2}\right)\)

Step-by-Step Solution

Answer

Straight line: \( \sqrt{2} \); Quarter-circle: \( \frac{1}{2} \pi \sqrt{2} \); Semicircle: \( \pi \times \frac{\sqrt{2}}{2} \).

Step 1: Calculate Straight Line Distance

To find the distance between the points (1,0) and (0,1) along a straight line, we use the distance formula. The distance formula is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substituting the coordinates of the points gives: \( d = \sqrt{(0 - 1)^2 + (1 - 0)^2} = \sqrt{1 + 1} = \sqrt{2} \).

Step 2: Determine Quarter-Circle Arc Length

For a quarter-circle path from (1,0) to (0,1) centered at (0,0), we calculate 1/4th the circumference of a circle with radius \( r = \sqrt{2} \) (since each point is \( \sqrt{2}/2 \) away from the origin). The formula for the arc length is \( L = \frac{1}{4} \times 2\pi \times r = \frac{1}{2} \pi r \). Substituting \( r = \sqrt{2} \), we find \( L = \frac{1}{2} \pi \sqrt{2} \).

Step 3: Compute Semicircle Arc Length

For a semicircle between (1,0) and (0,1) centered at \((\frac{1}{2}, \frac{1}{2})\), the radius \( r \) is \( \frac{\sqrt{2}}{2} \). The length of the semicircular path is \( L = \pi r \). Substituting \( r = \frac{\sqrt{2}}{2} \), we have \( L = \pi \times \frac{\sqrt{2}}{2} \).

Key Concepts

Straight Line Distance

To calculate the straight line distance between the points (1, 0) and (0, 1), we use the well-known distance formula. This formula helps us find the shortest path connecting two points in a plane, easily calculated using their coordinates. The distance formula is given by:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]where

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
• The squared differences find changes along each axis.
• The square root gives us the actual distance.

Substituting the given points \((1, 0)\) and \((0, 1)\) results in: \[ d = \sqrt{(0 - 1)^2 + (1 - 0)^2} = \sqrt{1 + 1} = \sqrt{2} \]. This formula is vital because it measures the most direct route—like a ruler tracing a line between the points.

Arc Length

Arc length calculation is essential when determining the distance along curved paths. Suppose we have a quarter-circle arc from (1, 0) to (0, 1), centered at (0,0). In that case, this path is part of the circumference of a circle.To find the arc length for a circle, we first calculate the circle's full circumference. With the circle's radius \(r\) known, the equation is: \[ C = 2 \pi r \] Here, since it's a quarter-circle, we take a fraction of the entire circumference:\[ L = \frac{1}{4} \times 2 \pi \times r = \frac{1}{2} \pi r \]In the provided exercise, the radius \(r = \sqrt{2}\). By plugging the radius into the arc length formula, we find \( L = \frac{1}{2} \pi \sqrt{2} \). Understanding this concept is important because it shows how parts of circlesrelate to linear measurements.

Semicircle

A semicircle is half of a full circle. In terms of distance, the semicircular path between two points encompasses the half-circumference of a circle.The given problem requires us to find the semicircle's arc length between (1,0) and (0,1) with the center at \((\frac{1}{2}, \frac{1}{2})\).To tackle this, identify the circle's radius first, given here as \( r = \frac{\sqrt{2}}{2} \). Since we seek the distance along a semicircle, our arc length formula modifies to:\[L = \pi r \]Thus, substituting the radius in this case,\( L = \pi \times \frac{\sqrt{2}}{2} \). Recognizing what a semicircle entails helps in visualizing the influence of curves on distance calculations. With fine-tuning of calculations and learning about partial circumferences, you can grasp the essentials of curved geometrical paths.