# 3.9 – Functions and Formulas

Objectives

• 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 $d=rt$)
• Rate is the same as Speed
• Distance for Free-Falling Objects: $d=(4.9)t^2$

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: $T(w)=10+0.5(\frac{w-100}{10})$
• 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). • $T(130)=10+0.5(\frac{130-100}{10})$ • $T(130)=10+0.5(\frac{30}{10})$ • $T(130)=10+0.5(3)$ • $T(130)=10+1.5$ • $T(130)=11.5$ • 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
• $T(60)=\frac{720}{60}$
• $T(60)=12$
• 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
• $T(1300)=\sqrt{\frac{1300}{4.9}}$
• $T(1300)=\sqrt{265.31}$
• $T(1300)=16.3$
• The time it takes for your phone to be utterly destroyed falling off the Freedom Tower is 16.3 seconds.