Problem 73 - College Algebra Essentials

Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$\frac{1}{p}+\frac{1}{q}=\frac{1}{f} \text { for } f$$

Step-by-Step Solution

Answer

The formula solved for \( f \) is \( f = \frac{p \times q}{p + q} \). This is the lens/mirror formula in physics, which describes the relationship between the object distance (p), image distance (q), and the focal length (f) of a lens or mirror.

Step 1: Identify the variable to solve for

The given formula is \( \frac{1}{p}+\frac{1}{q}=\frac{1}{f} \). This needs to be solved for \( f \).

Step 2: Rewrite the formula

To solve for \( f \), you first need to get \( f \) by itself on one side of the equation. Multiply each term by \( p \times q \times f \) to do this. This results in \( q + p = f \times p \times q \).

Step 3: Isolate \( f \)

To isolate \( f \), divide each side by \( p \times q \), which results in the equation \( f = \frac{p \times q}{p + q} \).

Step 4: Identify the meaning of the formula

Upon successful extraction of \( f \) in the equation, it becomes clear that this formula is known as the lens/mirror equation in physics which describes the relationship between the object distance (p), image distance (q), and the focal length (f) of a lens or mirror. Note that p and q are measured from the lens/mirror to the object and image respectively, and f is the focal length.