Problem 83 - Precalculus: Functions and Graphs

Question

Many calculators have viewing sereens that are wider than they are high. The approximate ratio of the height to the width is often \(2: 3 .\) Let the actual height of the calculator screen along the \(y\) -axis be 2 units, the actual width of the calculator screen along the \(x\) -axis be 3 units, and Xscl \(=\mathbf{Y s c}=1 .\) since the line \(y=x\) must pass through the point \((1,1),\) the actual slope \(m_{A}\) of this line on the calcuIator screen is given by $$m_{A}=\frac{\text { actual distance between tick marks on } y \text { -axis }}{\text { actual distance between tick marks on } x \text { -axis }}$$ Using this information, graph \(y=x\) in the given viewing rectangle and predict the actual angle \(\boldsymbol{\theta}\) that the graph makes with the \(x\) -axis on the viewing screen. \([0,3]\) by \([0,2]\)

Step-by-Step Solution

Answer

The angle \( \theta \) is approximately \( 33.69^\circ \).

Step 1: Understanding the Problem

We need to find the actual angle \( \theta \) that the line \( y = x \) makes with the \( x \)-axis on the calculator screen, given the viewing rectangle with width 3 units and height 2 units.

Step 2: Finding the Actual Slope

The slope \( m = \frac{\Delta y}{\Delta x} \) of the line \( y = x \) in a typical coordinate system is 1. However, on the screen, we need to consider the actual distances between tick marks: \( m_A = \frac{1}{1} = 1 \), as the scales along both axes are 1.

Step 3: Adjusting for Screen Dimensions

The given problem states the actual physical screen has height 2 units and width 3 units, making the viewing rectangle non-square. Hence, the width to height ratio of tick marks is \( \frac{3}{2} \).

Step 4: Calculate the Actual Slope with Screen Ratios

Since the screen has a different height-to-width ratio of \( 2:3 \), the actual slope \( m_A \) on the screen is given by the proportion: \( m_A = \frac{2 \text{ units}}{3 \text{ units}} = \frac{2}{3} \).

Step 5: Find the Angle Using Slope

The angle \( \theta \) that the line makes with the \( x \)-axis is given by \( \theta = \arctan(m_A) \). Substitute the value of \( m_A \): \( \theta = \arctan\left(\frac{2}{3}\right) \).

Step 6: Calculate the Actual Angle \(\theta\)

Use a calculator to find \( \theta = \arctan\left(\frac{2}{3}\right) \approx 33.69^\circ \).

Key Concepts

Slope

In coordinate geometry, the concept of the slope is fundamental for understanding how lines behave in a space. Slope describes how steep a line is by quantifying how much it goes up or down for a given horizontal distance. It is typically given by the formula:

• \( m = \frac{\Delta y}{\Delta x} \)

where \( \Delta y \) is the change in the vertical direction (along the \(y\)-axis) and \( \Delta x \) is the change in the horizontal direction (along the \(x\)-axis). In the case of the line \( y = x \) in a standard coordinate system, the slope is 1, since the line rises one unit for every unit it runs.

The exercise involves interpreting this simple slope equation within a realistic screen dimension context, where the screen's height and width affect the perception of the slope. Given the non-square aspect of the screen with a width-to-height ratio of 3:2, the slope must be recalculated as \( m_A = \frac{2}{3} \). This adjustment accounts for the screen's specific aspect ratio impacting how the slope is visualized.

Angle Calculation

Calculating the angle a line makes with the \(x\)-axis involves understanding how the slope translates into angular measurement. Once we have the slope value, we can calculate the angle using the arctangent function, a trigonometric function that returns the angle whose tangent is a given number. The angle \( \theta \) made by a line with the \(x\)-axis can be found with:

• \( \theta = \arctan(m) \)

For a line \( y = x \), in an ordinary unit-square system, this angle is \( 45^{\circ} \), as the slope is 1.

However, on the specified screen with a slope of \( m_A = \frac{2}{3} \), the angle needs recalibration. Using \( \theta = \arctan\left(\frac{2}{3}\right) \), which results in approximately \( 33.69^{\circ} \), we account for the screen's perspective in our calculation. This difference exemplifies how the angle is influenced by the coordinate system's scaling and physical dimensions, integral when plotting graphs on varied devices.

Aspect Ratio

Aspect ratio refers to the proportional relationship between a screen's width and height, which in this case is given by \( 3:2 \). It's an important concept in coordinate geometry, as it massively influences how visual information, like graphical lines, is perceived on different devices.

This particular exercise emphasizes the impact of aspect ratio on graphical representation by altering the line \( y = x \). While theoretically unchanged in its mathematical form, the screen's aspect ratio modifies the slope perceived on the device, leading to an apparent slope of \( m_A = \frac{2}{3} \). The notion of aspect ratio extends beyond this exercise; it applies broadly wherever graphics need to be accurately represented across diverse display formats.

• Understanding aspect ratios ensures that graphical data appears correctly across various media devices.
• Such understanding is crucial in fields like graphic design and multimedia display technology, which rely on precise visual scaling.

By bridging the relationship between coordinate geometry and physical display characteristics, this exercise hints at the underlying geometry that dictates how content is visually structured on different sized screens.