Problem 52 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. All numbers are accurate to at least two significant digits. When focusing a camera, the distance \(r\) the lens must move from the infinity setting is given by \(r=f^{2} /(p-f),\) where \(p\) is the distance from the object to the lens, and \(f\) is the focal length of the lens. Solve for \(f\).

Step-by-Step Solution

Answer

Use the quadratic formula to find \( f = \frac{-r + \sqrt{r^2 + 4rp}}{2} \).

Step 1: Objective of the Exercise

We need to solve the equation \( r = \frac{f^2}{p-f} \) for the variable \( f \).

Step 2: Identify and Simplify

Start by multiplying both sides of the equation by \( p-f \) to eliminate the fraction, giving us \( r(p-f) = f^2 \).

Step 3: Expand the Equation

Distribute \( r \) on the left side to get \( rp - rf = f^2 \).

Step 4: Rearrange the Equation

Move all terms involving \( f \) to one side. The equation becomes \( f^2 + rf = rp \).

Step 5: Apply Quadratic Formula

Recognize the equation in the form of a quadratic \( f^2 + rf - rp = 0 \). Use the quadratic formula \( f = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 1 \), \( b = r \), and \( c = -rp \).

Step 6: Substitute into the Quadratic Formula

Substitute the values into the quadratic formula:\[f = \frac{-r \pm \sqrt{r^2 + 4rp}}{2}\]

Step 7: Analyze Solutions

The quadratic formula provides two possible solutions for \( f \), but in a physical context, choose the positive solution as focal length must be positive:\[f = \frac{-r + \sqrt{r^2 + 4rp}}{2}\]

Key Concepts

Quadratic Formula

The quadratic formula is an essential mathematical tool used to find solutions to quadratic equations of the form \(ax^2 + bx + c = 0\). These equations arise in various problems, including the one at hand where we solve for the lens's focal length \(f\). The quadratic formula is: \[f = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Here, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation. In our problem, \(a = 1\), \(b = r\), and \(c = -rp\). By replacing these values into the formula, we calculate the possible values of \(f\). The beauty of the quadratic formula lies in its universality—it can quickly provide solutions for any quadratic equation by simply substituting in the appropriate coefficients. Understanding the discriminant \(b^2 - 4ac\) is vital. It helps determine the nature of the roots:

• If it's positive, there are two distinct real roots.
• If it's zero, there is one real root (a repeated root).
• If it's negative, the roots are complex and not real.

This aspect aids problem solvers in predicting the type of solutions without explicit calculation.

Algebraic Manipulation

Algebraic manipulation is a technique used to rearrange equations by applying mathematical operations. It simplifies the process of isolating a variable, as seen in this exercise to solve for \(f\). Initially, we began with the equation \(r = \frac{f^2}{p-f}\). By multiplying through by \(p-f\), we removed the fraction, resulting in \(r(p-f) = f^2\). This step is crucial because handling simpler forms of an equation eases further calculations. As we continue, we distribute the \(r\) to get \(rp - rf = f^2\). This expansion is a common algebraic technique to eliminate parentheses and simplify expressions. Once simplified, rearranging the terms, \(f^2 + rf = rp\), sets up the equation in a familiar quadratic form. Effective algebraic manipulation requires:

• Understanding equivalent transformations, which do not alter the equation's solutions.
• Knowing how to distribute factors correctly.
• Recognizing patterns that match known forms like quadratic equations.

These skills make algebraic manipulation a cornerstone of mathematical problem solving.

Mathematical Problem Solving

Mathematical problem solving involves applying mathematical concepts and procedures to find answers to questions or predictions about real-world scenarios. Here, we're tasked with solving for \(f\), the focal length of a lens, by using the relationship given in the exercise. The steps include:

• Understanding the problem statement and identifying what needs to be solved.
• Applying known formulas and techniques, such as the quadratic formula, to solve the problem.
• Using algebraic manipulation to simplify the steps leading up to the quadratic equation.
• Analyzing solutions to determine which is feasible; for instance, choosing the positive solution because focal length must be positive.

Effective problem solving in mathematics requires combining different strategies and using critical thinking to choose the best path to the solution. It also involves verifying results, ensuring they make sense contextually, as demonstrated by opting for the positive root of the quadratic solution, which aligns with physical realities. By approaching problems systematically, one can tackle a variety of mathematical challenges confidently.