91Ó°ÊÓ

In mathematics, direct variation describes a linear relationship between two variables where one variable is a constant multiple of the other. This is expressed in the form \( y = kx \), where \( y \) varies directly as \( x \), and \( k \) is the constant of variation. If \( x \) increases, \( y \) increases proportionally, which means they either both rise or fall together.

For example, in the context of photography and the given problem, the f-number of a lens varies directly with the square root of the time that the film is exposed to light. This can be captured by the equation \( f = k \sqrt{t} \). It signifies that if the exposure time increases, the f-number increases in proportion to the square root of the time. This relationship helps photographers determine the appropriate f-number needed for different exposure times.

• Direct variation involves linear relationships.
• This concept supports calculations in real-life applications, like determining lens settings in photography.

Understanding how direct variation works is crucial in fields that require precise measurements and adjustments, such as photography, engineering, and even finance.

F-number, often seen in the format \( f/x \), is a critical concept in photography that influences image brightness and depth of field. It is determined by the formula \( f = k \sqrt{t} \) in scenarios of direct variation involving time \( t \) as in the given problem.
This formula not only helps compute the f-number when given the time but also enables you to predict how a change in exposure time will affect the f-number. In the exercise, for an initial time of 0.0200 seconds with an f-number of 8, the constant \( k \) is computed first. Then, this constant allows calculation of the new f-number for a different time, 0.0098 seconds.

The key steps in calculating the f-number:

• Identify the exposure time \( t \) and corresponding f-number.
• Solve for the constant \( k \).
• Use \( k \) to find the f-number for any other given time \( t \).

These calculations help photographers adjust camera settings to match varying lighting conditions without compromising on image quality.

The constant of variation, denoted as \( k \), remains consistent in a direct variation problem and is essential for converting an equation into a practical tool for solving problems.

In our camera setting problem, \( k \) was determined with known values of the f-number and exposure time. Once calculated, \( k \) is used to find the f-number for any other exposure time by substituting back into the equation \( f = k \sqrt{t} \).
By solving \( 8 = k \sqrt{0.0200} \) and finding \( k \approx 56.56 \), the equation confidently adjusts to find an accurate f-number for a different time. This approach exemplifies the power of constant \( k \) in applying theoretical mathematics to practical scenarios.

• Constant \( k \) links the mathematical model to real-world application.
• Knowing \( k \) allows seamless transitions between different scenarios.

Once understood, the constant of variation becomes a versatile figure in various scientific and artistic fields, providing a bridge between theory and application.