Problem 29 - Precalculus: Functions and Graphs

Question

Density at a point \(A\) thin flat plate is situated in an \(x y\) -plane such that the density \(d\) (in Ib/ft \(^{2}\) ) at the point \(P(x, y)\) is inversely proportional to the square of the distance from the origin. What is the effect on the density at \(P\) if the \(x\) - and \(y\) -coordinates are each multiplied by \(\frac{1}{3} ?\)

Step-by-Step Solution

Answer

The density increases by a factor of 9.

Step 1: Understand the Proportional Relationship

The density \(d\) at a point \(P(x, y)\) is given as inversely proportional to the square of the distance from the origin \((0, 0)\). This means we can express it as \(d = \frac{k}{r^2}\), where \(k\) is a constant and \(r\) is the distance from the origin.

Step 2: Find the Expression for Distance

The distance \(r\) from the origin to the point \(P(x, y)\) is calculated using the distance formula: \(r = \sqrt{x^2 + y^2}\). Therefore, the density \(d\) becomes \(d = \frac{k}{(x^2 + y^2)}\).

Step 3: Transform Coordinates and Calculate New Distance

If the \(x\)- and \(y\)-coordinates are each multiplied by \(\frac{1}{3}\), the new point is \((\frac{x}{3}, \frac{y}{3})\). The new distance from the origin is \(r' = \sqrt{(\frac{x}{3})^2 + (\frac{y}{3})^2}\).

Step 4: Simplify the New Distance Expression

Calculate \(r'\) as \(r' = \sqrt{\frac{x^2}{9} + \frac{y^2}{9}} = \frac{1}{3} \sqrt{x^2 + y^2}\). Thus, \(r' = \frac{r}{3}\).

Step 5: Calculate the New Density

The new density \(d'\) is \(d' = \frac{k}{(r')^2} = \frac{k}{(\frac{r}{3})^2} = \frac{k}{\frac{r^2}{9}} = 9 \times \frac{k}{r^2}\). Therefore, \(d' = 9d\).

Step 6: Conclude the Effect on Density

When the \(x\) and \(y\) coordinates are each divided by 3, the density at point \(P\) becomes 9 times its original value.

Key Concepts

Density Function

The concept of a density function is essential in understanding how quantities like mass or charge distribute over a given area. In the scenario of the thin flat plate in the xy-plane, the density at any point is not constant. Instead, it changes based on its distance from the origin. This relationship is expressed using the equation \(d = \frac{k}{r^2}\), where \(k\) is a constant and \(r\) is the distance from the origin. The key feature of this equation is its inverse proportionality to \(r^2\), meaning that as the distance \(r\) increases, the density \(d\) decreases, and vice versa.

• Inverse proportionality indicates that the density decreases as the distance from the origin increases.
• The formula \(d = \frac{k}{r^2}\) provides a mathematical model to represent this proportionality.

Understanding this helps us predict how the density will change if we alter the position of the point \(P(x, y)\).

Distance Formula

To calculate the density at any point on the plane, we first need the distance from the origin to that point. The distance formula is derived from the Pythagorean theorem and is given as \(r = \sqrt{x^2 + y^2}\). This equation allows us to find the straight-line distance from a central point, here it's the origin \((0,0)\), to any other point \((x,y)\) on the plane.

• This formula is critical to calculating the change in density as it directly affects the inverse proportional relationship.
• Substituting \(r\) in the density equation \(d = \frac{k}{r^2}\) shows the dependency of density on distance.

Thus, finding \(r\) accurately using this formula is key to understanding and manipulating density values at various points.

Coordinate Transformation

Coordinate transformation involves modifying the \(x\) and \(y\) values of a point \((x,y)\). In the provided exercise, both the \(x\) and \(y\) coordinates of any point are multiplied by \(\frac{1}{3}\). This results in a new point \((\frac{x}{3}, \frac{y}{3})\).

• Such a transformation affects the overall distance from the origin because the point effectively gets closer to it.
• This reducing effect on the coordinate values influences the resulting density due to the unchanged relationship of \(d = \frac{k}{r^2}\).

By understanding how transformations work, we can predict changes in properties like density without requiring specific knowledge of each transformation point's initial location.

Effect on Density

When the coordinates of a point are scaled by a factor, it directly impacts the density due to the inverse proportionality to distance squared. In this problem, when both \(x\) and \(y\) coordinates are divided by 3, each transformation effectively scales the original distance by \(\frac{1}{3}\), leading the new distance \(r'\) to be \(\frac{r}{3}\). Consequently, substituting into the density formula gives the new density as \(d' = 9d\).

• This means the density becomes 9 times greater because distance's contribution to inverse proportion decreases due to reduced distance.
• Such changes illustrate how sensitive density is to changes in positioning and showcase the power of transformation in manipulating physical properties.

Thus, understanding these effects allows us to predict and control the density alterations without additional material changes.