Problem 226 - Calculus Volume 1

Question

A local art gallery has a portrait \(3 \mathrm{ft}\) in height that is hung \(2.5 \mathrm{ft}\) above the eye level of an average person. The viewing angle \(\theta\) can be modeled by the function \(\theta=\tan ^{-1} \frac{5.5}{x}-\tan ^{-1} \frac{2.5}{x},\) where \(x\) is the distance (in feet) from the portrait. Find the viewing angle when a person is \(4 \mathrm{ft}\) from the portrait.

Step-by-Step Solution

Answer

The viewing angle is approximately 0.386 radians.

Step 1: Identify the Function

We are given the function for the viewing angle as follows: \[ \theta = \tan^{-1} \left( \frac{5.5}{x} \right) - \tan^{-1} \left( \frac{2.5}{x} \right) \]where \(x\) is the distance from the portrait.

Step 2: Substitute the Distance

Next, we substitute \(x = 4\) feet into the function:\[ \theta = \tan^{-1} \left( \frac{5.5}{4} \right) - \tan^{-1} \left( \frac{2.5}{4} \right) \]

Step 3: Calculate the Arctangents

Calculate each arctangent separately:1. First calculate \( \tan^{-1} \left( \frac{5.5}{4} \right) \).2. Then calculate \( \tan^{-1} \left( \frac{2.5}{4} \right) \).

Step 4: Evaluate the Expression

Evaluate both arctangents and subtract the second from the first to find \( \theta \):- Let's say \( \tan^{-1} \left( \frac{5.5}{4} \right) \approx 0.944\) radians.- And \( \tan^{-1} \left( \frac{2.5}{4} \right) \approx 0.558\) radians.- Therefore, \( \theta \approx 0.944 - 0.558 = 0.386\) radians.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when given a ratio of sides in a triangle. Imagine you have a right triangle and know the lengths of two sides. You want to determine the measure of one of the non-right angles. That's where inverse trigonometric functions step in.
In this context, we have arctangent, denoted as \( \tan^{-1} \). It's specifically used when you know the opposite and adjacent sides of a right triangle. By applying the inverse tangent function, you can calculate the angle corresponding to these sides.

• \( \tan^{-1} \left( \frac{5.5}{x} \right) \) gives you the angle opposite to the side 5.5 feet.
• \( \tan^{-1} \left( \frac{2.5}{x} \right) \) provides the angle opposite to the side 2.5 feet.

This makes inverse trigonometric functions powerful tools in geometry and helps us determine angles critical in many mathematical applications.

Viewing Angle

The viewing angle in our problem is the angle formed by your line of sight from your eyes to the top and bottom of the portrait. It changes based on your distance from the portrait, a factor represented by \( x \) in the formula.
The problem uses the formula \( \theta = \tan^{-1} \left( \frac{5.5}{x} \right) - \tan^{-1} \left( \frac{2.5}{x} \right) \) to express this concept. This equation calculates the portion of your view taken up by the portrait as seen from a specific distance.

• A wider viewing angle means the portrait appears larger within your line of sight.
• A smaller viewing angle implies the portrait takes up less of your view.

The ideal viewing angle influences how art is displayed in galleries, ensuring clarity and appeal.

Arctangent

The function \( \tan^{-1} \), or arctangent, is crucial in determining angles when we have a relation between two sides of a triangle. It's the inverse of the tangent function, which helps find an angle given its tangent value.
In the exercise, arctangent functions handle the comparison of heights and distances to solve for the optimal viewing angle. To find the viewing angle when you're 4 feet away from the portrait:

• First, calculate \( \tan^{-1} \left( \frac{5.5}{4} \right) \) which results in an angle of approximately 0.944 radians.
• Next, determine \( \tan^{-1} \left( \frac{2.5}{4} \right) \) giving an angle of about 0.558 radians.
• Finally, subtract these two angles to determine the viewing angle, which is \( 0.944 - 0.558 = 0.386 \) radians.

Arctangent provides a straightforward method to find the critical angle needed to appreciate the portrait fully.