Ideal Gas Law (Chemistry/Physics)

Equation:

where

• is the pressure,
• is the volume,
• is the number of moles of gas,
• is the universal gas constant,
• is the temperature.

Goal: Solve for in terms of , , , and .

Solution

- Since the equation is linear in , isolate by dividing both sides by :

Perimeter of a Rectangle (Everyday Geometry)

Equation:

where

• is the perimeter of a rectangle,
• is the length,
• is the width.

Goal: Solve for in terms of and .

Solution

- Subtract from both sides: - Divide both sides by to isolate :

Projectile Motion (Physics/Engineering)

Equation (vertical position):

where

• is the time (in seconds),
• is the initial velocity,
• is the initial height,
• is the constant reflecting gravitational acceleration (ft/s).

Goal: Solve for in terms of , , and .

Solution

- Rewrite the equation to set it equal to zero: - Use the quadratic formula , letting , , and : - Simplify:

Lens Formula (Optics)

Equation:

where

• is the focal length of the lens,
• is the distance from the lens to the object,
• is the distance from the lens to the image.

Goal: Solve for in terms of and .

Solution

- Isolate : - Combine the right-hand side: - Therefore: - Take the reciprocal:

Period of a Pendulum (Physics)

The period of a simple pendulum is approximately given by

where

• is the period (in seconds),
• is the length of the pendulum (in meters),
• is the acceleration due to gravity (in m/s).

Goals:

• (A) Solve for in terms of and .
• (B) Solve for in terms of and .

Solution

- **Goal A:** Square both sides and solve for . - **Goal B:** From , rearrange to solve for .

Pythagorean Theorem (Geometry/Engineering)

Equation:

where

• and are the legs of a right triangle,
• is the hypotenuse.

Goal: Solve for in terms of and .

Solution

- Subtract from both sides: - Take the positive square root (assuming ):