Lesson 6.6 Radical Equations

Transcription

1 Lesson 6.6 Radical Equations Activity 1 Radical Equations 1. a. To solve a radical equation, we: (1) isolate the radical, then () square both sides. Be careful when squaring a binomial! For example, # # ÐB %Ñ œ ÐB %ÑÐB %Ñ œ B )B "' It is not true that # # # # ÐB %Ñ œ B % œ B "' WRONG! FOIL it out Square some binomials. Try to do the calculation mentally. ÐB &Ñ # œ ÐA 'Ñ # œ Ð#> $Ñ # œ Ð$@ %Ñ # œ b. Follow the steps to solve the equation #B $ œ È%B $ Square both sides. Get zero on one side. Solve the quadratic equation. Check both solutions: B œ $ À B œ " À You should find that one of the "solutions" checks, and the other doesn't. Why does this happen? 13

2 c. Use a graph to solve the equation #B $ œ È%B $ Graph Y " œ #B $ and Y # œ È%B $ on the grid below left. How many solutions? # Graph Y " œ Ð#B $Ñ and Y # œ %B $ on the grid above right. How many solutions? What happened when we squared both sides of the equation?. If a solution doesn't check, it is a false or extraneous solution. Whenever we square both sides of an equation, we must always check for extraneous solutions! Sometimes we have to square both sides twice in order to get rid of all the radicals. Follow the steps to solve the equation È È C % œ C #! # Isolate the more complicated radical. Square both sides. Isolate the term with the radical. Divide both sides by %. Square both sides again. Solve for C. Check the solution(s). Solution: 133

3 Activity 1. Simplify: Roots and Powers are Opposite Operations a. È *# È # œ Ð *Ñ œ È * È * œ b. È &# È # œ Ð &Ñ œ È & È & œ c. È$ ) $ È$ $ œ Ð )Ñ œ È$ ) È$ ) È$ ) œ d. È$ % $ È$ $ œ Ð %Ñ œ È$ % È$ % È$ % œ. Did you use your calculator? You don't need it! Explain why not. 3. Simplify. (Assume that B is a positive number.) a. È B# È # œ Ð BÑ œ ÈBÈB œ b. È$ B$ È $ $ œ Ð BÑ œ È $ BÈ $ BÈ $ B œ Activity 3 Kleiber's Rule Perhaps the single most useful piece of information a scientist can have about an animal is its metabolic rate. The resting or basal metabolic rate, BMR, is the minimum amount of energy the animal can expend in order to survive. Kleiber's rule states that the BMR for many groups of animals is given by FÐ7Ñ œ (!7 $Î% where 7 is the animal's mass in kilograms, and the BMR is measured in kilocalories per day. a. Calculate the BMR for various animals whose masses are given in the table. Animal Bat Squirrel Raccoon Lynx Human Moose Rhinoceros Weight (kg)!þ"!þ' ) $! (! $'! $&!! BMR (kcalîday) 13

4 b. Plot the data and sketch a graph of Kleiber's rule for! 7 Ÿ %!!. c. Write and solve an equation to answer the question: What is the mass of an animal whose BMR is 7500 kcal Î day? d. A more accurate version of Kleiber's rule for mammals is VÐ7Ñ œ ($Þ$7!Þ(% Use this formula to answer the question in part (c). Activity The Gossamer Albatross A bicycle ergometer is used to measure the amount of power generated by a cyclist. The scatterplot shows how long an athlete was able to sustain various levels of power output. The curve is the graph of C œ &!!B!Þ#* which approximately models the data. a. In this graph, which variable is the input and which is the output? b. The athlete maintained 650 watts of power for 0 seconds. What power output does the equation predict for 0 seconds? 135

5 c. The athlete maintained 300 watts of power for 10 minutes. How long does the equation predict that power output can be maintained? d. In 1979 a remarkable pedal-powered aircraft called the Gossamer Albatross was successfully flown across the English Channel. The flight took 3 hours. According to the equation, what level of power can be maintained for 3 hours? e. The Gossamer Albatross needed 50 watts of power to keep it airborne. For how long can 50 watts be maintained, according to the given equation? Wrap-Up In this Lesson, we worked on the following skills and goals related to powers and roots: Solving radical equations Checking for extraneous solutions Solving formulas involving radicals Simplifying roots of powers Check Your Understanding 1. What is the most important thing to remember when solving a radical equation?. When can extraneous solutions occur? Why? 3. Explain why it is not always true that È B # œ B.. Explain how to solve the equation %B % œ È$B % by graphing. 136