Question

Photography The length \(L\) of the chord joining the endpoints of an arc on a circle of radius \(r\) subtended by a central angle \(\theta, 0<\theta \leq \pi,\) is given by \(L=r \sqrt{2-2 \cos \theta} .\) Use this fact to approximate the field width (the width of scenery the lens can image) of a \(450 \mathrm{~mm}\) camera lens at a distance of 920 feet if the viewing angle of the lens is \(\frac{\pi}{30}\).

Short answer

Approximately 80.38 feet.

Step-by-step solution

Identify Given Values

The radius of the circle is the distance from the lens to the object: \[ r = 920 \text{ feet} \] The central angle is: \[ \theta = \frac{\pi}{30} \] The formula for the chord length is: \[ L = r \sqrt{2 - 2 \cos \theta} \]

Substitute Values into the Formula

Substitute the given values for \( r \) and \( \theta \) into the chord length formula: \[ L = 920 \sqrt{2 - 2 \cos \left(\frac{\pi}{30}\right)} \]

Calculate \( \cos \left(\frac{\pi}{30}\right)\)

Find the cosine of the central angle: \[ \cos \left(\frac{\pi}{30}\right) \approx 0.99619 \]

Simplify the Expression Inside the Square Root

\[ 2 - 2 \cos \left(\frac{\pi}{30}\right) \approx 2 - 2(0.99619) \approx 2 - 1.99238 \approx 0.00762 \]

Compute the Square Root

\[ \sqrt{0.00762} \approx 0.08737 \]

Calculate the Chord Length

Multiply the result by the radius \( r \): \[ L = 920 \times 0.08737 \approx 80.38 \text{ feet} \]

Key concepts

central angle

In the context of a circle, a central angle is an angle whose vertex is at the center of the circle. The arms of this angle extend to the circumference and subtend an arc. The size of the central angle directly influences the size of this arc. In our exercise, the central angle is given as \(\frac{\text{π}}{30}\), which is measured in radians. Radians are a way of measuring angles based on the radius of the circle. When you have the central angle, you can calculate other properties of the circle, such as the length of the chord connecting the endpoints of the arc subtended by this angle.

radius of a circle

The radius of a circle is the distance from its center to any point on its circumference. This is a crucial measurement, as it helps determine many other properties of the circle, such as its area, circumference, and in this case, the chord length. In our exercise, the radius is 920 feet, which is the distance from the camera lens to the object being photographed. By knowing the radius and the central angle, we can use these values in geometric formulas like the chord length formula.

cosine function

The cosine function, often abbreviated as 'cos', is one of the primary trigonometric functions. It relates the angle of a right triangle to the lengths of its sides. In our exercise, the cosine function is used to calculate the chord length of a circle. Specifically, we need to find the cosine of the central angle \(\frac{\text{π}}{30}\). \(\text{cos}\frac{\text{π}}{30} \approx 0.99619\). The value of the cosine function helps us determine how much of the circle's radius contributes to the length of the chord. This function is essential when working with angles and distances in circles.

photographic field width

Photographic field width refers to the width of the scene that a camera lens can capture at a given distance. It can also be thought of as the width of the viewable area in an image. In this exercise, we approximate the photographic field width using the chord length formula. By knowing the lens's viewing angle \(\frac{\text{π}}{30}\) and the distance from the lens to the object (920 feet), we use the formula for chord length: \ L = r \sqrt{2 - 2 \text{cos} \theta} \. With this, we calculate the chord length to be approximately 80.38 feet, which represents the field width the camera lens can capture at that distance.