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3.9 – Functions and Formulas

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 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:

Alg1A 3.09 - ShippingTax

  • 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

Alg1A 3.09 - DistanceFunctionSpeed

    • 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

Alg1A 3.09 - HeightTimeDistance

    • 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.

 

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