Problem 29 - Calculus an Applied Approach

Question

In Exercises \(29-32,\) find the minimum distance from the curve or surface to the given point. (Hint: Start by minimizing the square of the distance.) $$ \begin{array}{l}{\text { Line: } x+y=6,(0,0)} \\ {\text { Minimize } d^{2}=x^{2}+y^{2}}\end{array} $$

Step-by-Step Solution

Answer

The minimum distance from the origin to the line x + y = 6 is \(3\sqrt{2}\) units.

Step 1: Understanding the Equation of the Line

Here, the equation of the line is given by x + y = 6. We can rewrite this equation in the standard form Ax + By + C = 0, where A = 1, B = -1, and C = -6.

Step 2: Using the Distance Formula

We will use the formula for the distance \(d\) between a point \((0,0)\) and the line \(Ax + By + C = 0\), which is given by: \[ d = \frac{|A(0) + B(0) + C|}{\sqrt{A^2 + B^2}}\]. Substituting the values of A, B, and C from step 1 into the formula, we get: \[ d = \frac{|-6|}{\sqrt{1^2 + (-1)^2}} = \frac{6}{\sqrt{2}}\]. Therefore, the required minimum distance is \(d = \frac{6}{\sqrt{2}}\).

Step 3: Simplifying the Result

We can simplify the denominator by the principle of rationalizing to get the distance in simplest form. When we rationalize \( \frac{6}{\sqrt{2}} \), we get \( \frac{6\sqrt{2}}{2} = 3\sqrt{2} \)

Key Concepts

Distance From Point to Line

Understanding the shortest distance from a point to a line is a common geometrical problem that has practical applications in various fields such as mapping and navigation. Given a point and a straight line, the minimum distance is the length of the perpendicular line segment from the point to the line.

To visualize this, imagine a vertical pole standing on a straight road; the shortest distance from the top of the pole to the road is a straight line dropping from the top to the road surface. In the context of the original exercise, the goal is to find the shortest distance from the origin, which is the point (0,0), to the line defined by the equation x + y = 6.

Distance Formula

The distance formula is fundamental when determining the distance between two points in the coordinate plane. Generally, this is calculated using the Pythagorean theorem in a Cartesian coordinate system which results in the formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

However, when it comes to finding the distance from a point to a line, we modify this approach. We use a specialized form of the distance formula: \( d = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} \) where \( Ax + By + C = 0 \) represents the equation of the line, and (x, y) is the point. This formula is derived from the equation of a line and the definition of perpendicular distance in Euclidean space.

Minimizing Square Distance

Minimizing the square of the distance, rather than the distance itself, is a strategy used in various optimization problems, including the one in our exercise. This technique is beneficial because it avoids dealing with square roots, which simplifies both the differentiation process and the algebra.

By minimizing \( d^2 \) instead of \( d \) itself, we bypass the need to calculate the derivative of a square root function when using calculus to find minimum values. Thus, we focus on minimizing \( x^2 + y^2 \) in the given problem to find the point on the line closest to the origin. The reason for squaring the distance is also connected to variance and standard deviation measures in statistics, where squaring distances from the mean mitigates the issue of dealing with negative values.

Equation of a Line

In algebra and geometry, the equation of a line is a fundamental concept, and it is typically expressed in the form \( Ax + By + C = 0 \) for a two-dimensional Cartesian plane. This equation relates all points \( (x, y) \) that lie on the line, with specific values for coefficients \( A \) and \( B \) indicating the slope and position of the line in the plane.

The equation provided in the exercise, \( x + y = 6 \) is a linear equation and can be rewritten in the standard form. By using this equation and known coordinates of a point, we can apply the distance formula to find the perpendicular and minimum distance from the point to the line, simplifying several types of geometrical analyses.