Work with a partner. a. Write a formula for the area A of a parallelogram.
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1 COMMON CORE Learning Standard HSA-CED.A.4 1. Rewriting Equations and Formulas Essential Question How can you use a formula for one measurement to write a formula for a different measurement? Using an Area Formula REASONING QUANTITATIVELY To e proficient in mat, you need to consider te given units. For instance, in Exploration 1, te area A is given in square inces and te eigt is given in inces. A unit analysis sows tat te units for te ase are also inces, wic makes sense. Work wit a partner. a. Write a formula for te area A of a parallelogram.. Sustitute te given values into te formula. Ten solve te equation for. Justify eac step. A = 30 in. 2 = in. c. Solve te formula in part (a) for witout first sustituting values into te formula. Justify eac step. d. Compare ow you solved te equations in parts () and (c). How are te processes similar? How are tey different? Using Area, Circumference, and Volume Formulas Work wit a partner. Write te indicated formula for eac figure. Ten write a new formula y solving for te variale wose value is not given. Use te new formula to find te value of te variale. a. Area A of a trapezoid. Circumference C of a circle 1 = 8 cm C = 24 π ft A = 63 cm 2 r 2 = 10 cm c. Volume V of a rectangular prism d. Volume V of a cone V = 7 yd 3 V = 24 π m 3 B = 12 π m 2 B = 1 yd 2 Communicate Your Answer 3. How can you use a formula for one measurement to write a formula for a different measurement? Give an example tat is different from tose given in Explorations 1 and 2. Section 1. Rewriting Equations and Formulas 3
2 1. Lesson Core Vocaulary literal equation, p. 36 formula, p. 37 Previous surface area Wat You Will Learn Rewrite literal equations. Rewrite and use formulas for area. Rewrite and use oter common formulas. Rewriting Literal Equations An equation tat as two or more variales is called a literal equation. To rewrite a literal equation, solve for one variale in terms of te oter variale(s). Rewriting a Literal Equation Solve te literal equation 3y + 4x = for y. 3y + 4x = 3y + 4x 4x = 4x 3y = 4x 3y 3 = 4x 3 Write te equation. Sutract 4x from eac side. Simplify. Divide eac side y 3. y = x Simplify. Te rewritten literal equation is y = x. Rewriting a Literal Equation Solve te literal equation y = 3x + xz for x. y = 3x + xz y = x(3 + z) Write te equation. Distriutive Property REMEMBER Division y 0 is undefined. y 3 + z = x(3 + z) 3 + z y 3 + z = x Simplify. y Te rewritten literal equation is x = 3 + z. Divide eac side y 3 + z. In Example 2, you must assume tat z 3 in order to divide y 3 + z. In general, if you ave to divide y a variale or variale expression wen solving a literal equation, you sould assume tat te variale or variale expression does not equal 0. Monitoring Progress Solve te literal equation for y. Help in Englis and Spanis at BigIdeasMat.com 1. 3y x = 2. 2x 2y = = 8x + 4y Solve te literal equation for x. 4. y = x 4x. 2x + kx = m x kx = y 36 Capter 1 Solving Linear Equations
3 Rewriting and Using Formulas for Area A formula sows ow one variale is related to one or more oter variales. A formula is a type of literal equation. Rewriting a Formula for Surface Area Te formula for te surface area S of a rectangular prism is S = 2 w w. Solve te formula for te lengt. w S = 2 w w S 2w = 2 w w 2w S 2w = 2 w + 2 S 2w = (2w + 2) S 2w 2w + 2 S 2w 2w + 2 = = (2w + 2) 2w + 2 Write te equation. Sutract 2w from eac side. Simplify. Distriutive Property Divide eac side y 2w + 2. Simplify. Wen you solve te formula for, you otain = S 2w 2w + 2. Using a Formula for Area You own a rectangular lot tat is 00 feet deep. It as an area of 100,000 square feet. To pay for a new water system, you are assessed $.0 per foot of lot frontage. a. Find te frontage of your lot.. How muc are you assessed for te new water system? w 00 ft frontage a. In te formula for te area of a rectangle, let te widt w represent te lot frontage. A = w Write te formula for area of a rectangle. A = w Divide eac side y to solve for w. 100,000 = w Sustitute 100,000 for A and 00 for = w Simplify. Te frontage of your lot is 200 feet.. Eac foot of frontage costs $.0, and $.0 1 ft 200 ft = $1100. So, your total assessment is $1100. Monitoring Progress Help in Englis and Spanis at BigIdeasMat.com Solve te formula for te indicated variale. 7. Area of a triangle: A = 1 ; Solve for Surface area of a cone: S = πr 2 + πr ; Solve for. Section 1. Rewriting Equations and Formulas 37
4 Rewriting and Using Oter Common Formulas Core Concept Common Formulas Temperature F = degrees Fareneit, C = degrees Celsius Simple Interest Distance C = (F 32) I = interest, P = principal, r = annual interest rate (decimal form), t = time (years) I = Prt d = distance traveled, r = rate, t = time d = rt Solve te temperature formula for F. Rewriting te Formula for Temperature C = (F 32) Write te temperature formula. C = F 32 Multiply eac side y. C + 32 = F Add 32 to eac side. C + 32 = F Simplify. Te rewritten formula is F = C Using te Formula for Temperature Wic as te greater surface temperature: Mercury or Venus? Convert te Celsius temperature of Mercury to degrees Fareneit. Mercury 427 C F = C + 32 Write te rewritten formula from Example. = (427) + 32 Sustitute 427 for C. Venus 864 F = Simplify. Because 864 F is greater tan F, Venus as te greater surface temperature. Monitoring Progress Help in Englis and Spanis at BigIdeasMat.com. A fever is generally considered to e a ody temperature greater tan 100 F. Your friend as a temperature of 37 C. Does your friend ave a fever? 38 Capter 1 Solving Linear Equations
5 Using te Formula for Simple Interest You deposit $000 in an account tat earns simple interest. After 6 monts, te account earns $162.0 in interest. Wat is te annual interest rate? COMMON ERROR Te unit of t is years. Be sure to convert monts to years. 1 yr 6 mo = 0. yr 12 mo To find te annual interest rate, solve te simple interest formula for r. I = Prt Write te simple interest formula. I = r Divide eac side y Pt to solve for r. Pt = r Sustitute for I, 000 for P, and 0. for t. (000)(0.) 0.06 = r Simplify. Te annual interest rate is 0.06, or 6.%. Solving a Real-Life Prolem A truck driver averages 60 miles per our wile delivering freigt to a customer. On te return trip, te driver averages 0 miles per our due to construction. Te total driving time is 6.6 ours. How long does eac trip take? Step 1 Rewrite te Distance Formula to write expressions tat represent te two trip times. Solving te formula d = rt for t, you otain t = d r. So, d 60 represents te delivery time, and d represents te return trip time. 0 Step 2 Use tese expressions and te total driving time to write and solve an equation to find te distance one way. d 60 + d = 6.6 Te sum of te two trip times is 6.6 ours. 0 11d = 6.6 Add te left side using te LCD d = 180 d = 180 Multiply eac side y 300 and simplify. Divide eac side y 11 and simplify. Te distance one way is 180 miles. Step 3 Use te expressions from Step 1 to find te two trip times. 60 mi So, te delivery takes 180 mi = 3 ours, and te return trip takes 1 0 mi 180 mi = 3.6 ours. 1 Monitoring Progress Help in Englis and Spanis at BigIdeasMat.com 10. How muc money must you deposit in a simple interest account to earn $00 in interest in years at 4% annual interest? 11. A truck driver averages 60 miles per our wile delivering freigt and 4 miles per our on te return trip. Te total driving time is 7 ours. How long does eac trip take? Section 1. Rewriting Equations and Formulas 3
6 1. Exercises Dynamic Solutions availale at BigIdeasMat.com Vocaulary and Core Concept Ceck 1. VOCABULARY Is r + 16 = π a literal equation? Explain. 2. DIFFERENT WORDS, SAME QUESTION Wic is different? Find ot answers. Solve 3x + 6y = 24 for x. Solve 24 3x = 6y for x. Solve 6y = 24 3x for y in terms of x. Solve 24 6y = 3x for x in terms of y. Monitoring Progress and Modeling wit Matematics In Exercises 3 12, solve te literal equation for y. (See Example 1.) 3. y 3x = x + y = 7. 2y 18x = x + y = 1 7. x y = y = 6. 4x = 7 + 4y x + = y 2x y = 3x y = 3 + 6x In Exercises 13 22, solve te literal equation for x. (See Example 2.) 13. y = 4x + 8x 14. m = 10x x 1. a = 2x + 6xz 16. y = 3x 7x 17. y = 4x + rx z = 8 + 6x px 1. sx + tx = r 20. a = x + cx + d x 4kx = y 22. x + 2wx = y 23. MODELING WITH MATHEMATICS Te total cost C (in dollars) to participate in a ski clu is given y te literal equation C = 8x + 60, were x is te numer of ski trips you take. a. Solve te equation for x.. How many ski trips do you take if you spend a total of $31? $48? 24. MODELING WITH MATHEMATICS Te penny size of a nail indicates te lengt of te nail. Te penny size d is given y te literal equation d = 4n 2, were n is te lengt (in inces) of te nail. a. Solve te equation for n.. Use te equation from part (a) to find te lengts of nails wit te following penny sizes: 3, 6, and 10. ERROR ANALYSIS In Exercises 2 and 26, descrie and correct te error in solving te equation for x x = 2(y x) 2x = 2(y x) 12 x = (y x) = ax 3 10 = x(a 3) 10 a 3 = x In Exercises 27 30, solve te formula for te indicated variale. (See Examples 3 and.) 27. Profit: P = R C; Solve for C. 28. Surface area of a cylinder: S = 2πr 2 + 2πr; Solve for. n 2. Area of a trapezoid: A = 1 2 ( ); Solve for Capter 1 Solving Linear Equations 30. Average acceleration of an oject: a = v 1 v 0 ; t Solve for v 1.
7 31. REWRITING A FORMULA A common statistic used in professional footall is te quarterack rating. Tis rating is made up of four major factors. One factor is te completion rating given y te formula R = ( C A 0.3 ) were C is te numer of completed passes and A is te numer of attempted passes. Solve te formula for C. 32. REWRITING A FORMULA Newton s law of gravitation is given y te formula F = G ( m 1 m 2 d 2 ) were F is te force etween two ojects of masses m 1 and m 2, G is te gravitational constant, and d is te distance etween te two ojects. Solve te formula for m PROBLEM SOLVING You deposit $2000 in an account tat earns simple interest at an annual rate of 4%. How long must you leave te money in te account to earn $00 in interest? (See Example 7.) 36. PROBLEM SOLVING A fligt averages 460 miles per our. Te return fligt averages 00 miles per our due to a tailwind. Te total flying time is 4.8 ours. How long is eac fligt? Explain. (See Example 8.) 37. USING STRUCTURE An atletic facility is uilding an indoor track. Te track is composed of a rectangle and two semicircles, as sown. x 33. MODELING WITH MATHEMATICS Te sale price S (in dollars) of an item is given y te formula S = L rl, were L is te list price (in dollars) and r is te discount rate (in decimal form). (See Examples 4 and 6.) r r a. Solve te formula for r.. Te list price of te sirt is $30. Wat is te discount rate? Sale price: $18 a. Write a formula for te perimeter of te indoor track.. Solve te formula for x. c. Te perimeter of te track is 660 feet, and r is 0 feet. Find x. Round your answer to te nearest foot. 34. MODELING WITH MATHEMATICS Te density d of a sustance is given y te formula d = m, were m is V its mass and V is its volume. 38. MODELING WITH MATHEMATICS Te distance d (in miles) you travel in a car is given y te two equations sown, were t is te time (in ours) and g is te numer of gallons of gasoline te car uses. Pyrite Density:.01g/cm 3 Volume: 1.2 cm 3 d = t d = 20g a. Solve te formula for m.. Find te mass of te pyrite sample. a. Write an equation tat relates g and t.. Solve te equation for g. c. You travel for 6 ours. How many gallons of gasoline does te car use? How far do you travel? Explain. Section 1. Rewriting Equations and Formulas 41
8 3. MODELING WITH MATHEMATICS One type of stone formation found in Carlsad Caverns in New Mexico is called a column. Tis cylindrical stone formation connects to te ceiling and te floor of a cave. stalagmite column a. Rewrite te formula for te circumference of a circle, so tat you can easily calculate te radius of a column given its circumference.. Wat is te radius (to te nearest tent of a foot) of a column tat as a circumference of 7 feet? 8 feet? feet? c. Explain ow you can find te area of a cross section of a column wen you know its circumference. 40. HOW DO YOU SEE IT? Te rectangular prism sown as ases wit equal side lengts. 41. MAKING AN ARGUMENT Your friend claims tat Termometer A displays a greater temperature tan Termometer B. Is your friend correct? Explain your reasoning. Termometer A F Termometer B 42. THOUGHT PROVOKING Give a possile value for. Justify your answer. Draw and lael te figure using your cosen value of. A = 40 cm 2 8 cm MATHEMATICAL CONNECTIONS In Exercises 43 and 44, write a formula for te area of te regular polygon. Solve te formula for te eigt center center a. Use te figure to write a formula for te surface area S of te rectangular prism.. Your teacer asks you to rewrite te formula y solving for one of te side lengts, or. Wic side lengt would you coose? Explain your reasoning. REASONING In Exercises 4 and 46, solve te literal equation for a. 4. x = a + + c a a 46. y = x ( a ) Maintaining Matematical Proficiency Evaluate te expression. (Skills Review Handook) Reviewing wat you learned in previous grades and lessons ( 6) + 3 Solve te equation. Grap te solutions, if possile. (Section 1.4) 1. x = 2. 3y 12 7 = r + 4 = s + = Capter 1 Solving Linear Equations
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