**Objectives**

- Identify formulas as ready-made equations that can help solve problems
- Change a general formula to fit the values of a specific problem
- Solve real-world problems using the distance formula and the formula for objects in free fall
- Solve a real-world business problem involving profit, revenue, and cost

**Key Terms**

- Formula – An equation that describes an important mathematical relationship.
- Input-Output Table – A two-column table whose first column contains input (independent, domain, x) values and whose second column contains output (dependent, range, y) values.
- Distance Rate Formula: Distance = Rate * Time (written as )
- Rate is the same as Speed

- Distance for Free-Falling Objects:

**Notes**

- Example: Tax Problem
- James wants to ship a box of springs from his company to another country.
- He knows that he must pay taxes.
- He looks at the sign:

- What do we know?
- The tax depends on the weight of the box: T(weight).
- The tax for 100 kg is $10.
- The tax for each additional 10 kg is $0.50.
- The box James is sending has a weight of 130 kilograms.
- Need to find T(130).

- Write the formula
- To find out the tax, break the 130 kilograms down into 100 kg + 30 kg.
- The first 100 kg costs $10, and each additional 10 kg costs $0.50 more.
- So:
- Because you have to pay $10 for 100 kg PLUS the charges ($0.50) for the rest of the weight (divided into 10 kg segments). The rest of the weight is: w minus the first 100 kg (which you already paid $10 for).
- The tax on 130 kg is $11.50.

- Example: Distance Rate Problem
- You are traveling 720 miles. How long will it take you to get to your destination if you travel at a rate (speed) of 60 miles per hour (mph)?
- What variable are you looking for?
- How long it will take: TIME

- What does TIME depend on?
- The speed of the car
- So, TIME(speed), T(s)

- You need to change the Distance Formula to solve for TIME

- Substitute the speed (60 mph) into the formula and solve for TIME
- The time it takes to drive 720 miles at 60 mph is 12 hours.

- Example: Distance Free-Falling Object Problem
- You are on the observation deck of the Freetom Tower in New York City (1,300 feet up from the street below).
- You accidentally drop your phone off the side while taking a photo.
- How long will it take your phone to crash into thousands of pieces on the street below (if it doesn’t hit anything or anyone on the way down)?
- What variable are you looking for?
- How long it will take: TIME

- What does TIME depend on?
- The height of the fall
- So, TIME(height), T(h)

- You need to change the Distance for Free-Falling Objects Formula to solve for TIME

- Substitute the height (1300 ft) into the formula and solve for TIME
- The time it takes for your phone to be utterly destroyed falling off the Freedom Tower is 16.3 seconds.