Problem 4 - Calculus Single Variable

Question

Which of the points \((2,1),(3,0),(4,-1),\) or \((1 / 2,9 / 2)\) is farthest from the origin? Which is nearest to the origin? Which is farthest from (-5,6)\(?\) Which is nearest to (10,7)\(?\)

Step-by-Step Solution

Answer

(4,-1) is farthest from the origin and (-5,6); nearest to the origin is (3,0); nearest to (10,7) are (2,1) and (4,-1).

Step 1: Understand the Distance Formula

The distance of a point \((x, y)\) from another point \((x_0, y_0)\) is given by the formula: \[ d = \sqrt{(x - x_0)^2 + (y - y_0)^2} \] We will use this formula to find the distances of each given point separately from the origin \((0,0)\), from \((-5,6)\), and from \((10,7)\).

Step 2: Calculate Distance from the Origin

For each point, calculate the distance to the origin using \[ d = \sqrt{x^2 + y^2} \]: - \((2, 1): d = \sqrt{2^2 + 1^2} = \sqrt{5}\) - \((3, 0): d = \sqrt{3^2 + 0^2} = 3\) - \((4, -1): d = \sqrt{4^2 + (-1)^2} = \sqrt{17}\) - \(\left(\frac{1}{2}, \frac{9}{2}\right): d = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{9}{2}\right)^2} = \frac{\sqrt{82}}{2}\)

Step 3: Determine Farthest from the Origin

Compare the calculated distances: \(\sqrt{5}\), \(3\), \(\sqrt{17}\), and \(\frac{\sqrt{82}}{2}\). Since \(\sqrt{17} \approx 4.123\), it's the largest. Thus, \((4, -1)\) is the farthest from the origin.

Step 4: Determine Nearest to the Origin

Compare again the calculated distances: - Smallest value is \(3\), which corresponds to the point \((3, 0)\). Thus, \((3, 0)\) is the nearest to the origin.

Step 5: Calculate Distance from (-5,6)

Calculate using \[ d = \sqrt{(x + 5)^2 + (y - 6)^2} \]: - \((2, 1): d = \sqrt{7^2 + (-5)^2} = \sqrt{74}\) - \((3, 0): d = \sqrt{8^2 + (-6)^2} = 10\) - \((4, -1): d = \sqrt{9^2 + (-7)^2} = \sqrt{130}\) - \(\left(\frac{1}{2}, \frac{9}{2}\right): d = \sqrt{\left(-\frac{9}{2}\right)^2 + \left(\frac{3}{2}\right)^2} = \frac{\sqrt{90}}{2}\)

Step 6: Determine Farthest from (-5,6)

Compare the calculated distances: - \(\sqrt{130}\) is the largest, which corresponds to \((4, -1)\). Thus, \((4, -1)\) is the farthest from \((-5, 6)\).

Step 7: Calculate Distance from (10,7)

Calculate using \[ d = \sqrt{(x - 10)^2 + (y - 7)^2} \]: - \((2, 1): d = \sqrt{(-8)^2 + (-6)^2} = 10\) - \((3, 0): d = \sqrt{(-7)^2 + (-7)^2} = \sqrt{98}\) - \((4, -1): d = \sqrt{(-6)^2 + (-8)^2} = 10\) - \(\left(\frac{1}{2}, \frac{9}{2}\right): d = \sqrt{-\frac{19}{2}^2 + -\frac{5}{2}^2} = \frac{\sqrt{410}}{2}\)

Step 8: Determine Nearest to (10,7)

Compare the calculated distances: - Smallest value is 10, which corresponds to both \((2, 1)\) and \((4, -1)\). Thus, \((2, 1)\) and \((4, -1)\) are both nearest to \((10, 7)\).

Key Concepts

Origin Distance

The distance of a point from the origin in a coordinate plane offers insight into its position relative to the center, which is located at \((0,0)\). To determine this distance for any point \((x, y)\), you apply the distance formula specifically tailored for the origin:

• \(d = \sqrt{x^2 + y^2}\)

This formula calculates the line segment, or the straight-line distance, between the origin and the point provided.
Applying it is manageable — just plug in the respective x and y values and solve.
For instance, if we have the point \((2, 1)\), we substitute and calculate \(\sqrt{2^2 + 1^2} = \sqrt{5}\).
The value represents the length from the origin to that point. In our solution, \((4, -1)\) came out farthest while \((3, 0)\) was closest.
By comparing these distances, we gain a clear picture of each point's proximity to the origin.

Coordinate Geometry

Coordinate geometry allows for the visualization of geometric principles on a grid using algebra. It brings to life geometric measurements through plotted points, lines, and figures using sets of numerical coordinates.
This system equips us to solve real-world geometric problems with efficiency. It is especially handy for calculating distances, angles, and slopes within a coordinate plane.

• The basic principle here deals with plotting points using x (horizontal) and y (vertical) coordinates.
• The Distance Formula is an algebraic approach that translates geometric properties into numerical calculations.
• This methodology enables identification of relationships between points without physical measurement.

The application to problems, such as identifying the proximity of points to specific locations, becomes seamless. Using coordinate geometry connecting algebra with geometry simplifies complex spatial inquiries.

Point Comparison

Point comparison is crucial for determining relative positions in coordinate geometry. Here, we compare given points based on their distances from specified points like the origin or certain coordinates such as \((-5,6)\) or \((10,7)\).
This comparison relies on calculated distances using the established formula.

• From our exercise, comparing points required calculating distances to evaluate which points were nearest or farthest.
• Distance comparisons allow clear conclusions on proximity, informing decisions or validating spatial relationships.

The specific approach of comparing points to each other aids in understanding geometric alignments and separation. This practice is fundamental in optimizing solutions in a multitude of contexts, ranging from mathematics to fields like robotics and logistics, where determining position effectively is crucial.