64 Vaiales and Fomulas Vaiales and Fomulas DEFINITIONS & BASICS 1) Vaiales: These symols, eing lettes, actually epesent numes, ut the numes can change fom time to time, o vay. Thus they ae called vaiales. Example: Tell me how fa you would e walking aound this ectangle. 24 ft 15 ft 15ft 24 ft It appeas that to get all the way aound it, we simply add up the numes on each side until we get all the way aound. 24+15+24+15 = 78. So if you walked aound a 24ft X 15ft ectangle, you would have completed a walk of 78 ft. I et we could come up with the patten fo how we would do this all of the time. Well, fist of all, we just pick geneal tems fo the sides of the ectangle: length width width Then we get something like this: length Distance aound the ectangle = length + width + length + width Let's ty and use some aeviations. Fist, peimete means aound measue. Sustitute it in: Peimete = length + width + length + width Let's go a it moe with just using the fist lettes of the wods: P = l + w + l + w Notice now how each lette stands fo a nume that we could use. The nume can change fom time to time. Since thei values can change, these lettes ae known as vaiales. The patten that we have ceated to descie all cases is called a fomula.
65 2) Fomula: These ae pattens in the fom of equations and vaiales, often with numes, which solve fo something we want to know, like the peimete equation efoe, o like: Aea of a ectangle: Volume of a Sphee: Pythagoean Theoem: A = B H = + = Because lettes ae used to epesent numes, we will see othe ways to wite multiplication:,, and putting things next to each othe. Like these: 5 7 = 35 5 7 = 35 5(7) = 35 Though the same pocess we can come up with many fomulas to use. Though it has all een made up efoe, thee is much to gain fom knowing whee a fomula comes fom and how to make them up on you own. I will show you on a couple of them. Distance, ate If you wee taveling at 40mph fo 2 hous, how fa would you have taveled? Well, most of you would e ale to say 80 mi. How did you come up with that? Multiplication: (40)(2) = 80 (ate of speed) (time) = distance o in othe wods: t = d whee is the ate t is the time d is the distance Pecentage If you ought something fo $5.50 and thee was an 8% sales tax, you would need to find 8% of $5.50 to find out how much tax you wee eing chaged..44 =.08(5.50) Amount of Tax = (inteest ate) (Puchase amount) o in othe wods: T = P Whee T is tax is ate of tax P is the puchase amount. Inteest This fomula is a summay of what we did in the last section with inteest. If you invested a pincipal amount of $500 at 9% inteest fo thee yeas, the amount in you account at the end of thee yeas would e given y the fomula: A = 500(1.09) 3 = $647.51
66 A = P(1 + ) Y whee A is the Amount in you account at the end P is the pincipal amount (stating amount) is the inteest ate Y is the nume of yeas that it is invested. Tempeatue Convesion Most of us know that thee is a diffeence etween Celsius and Fahenheit degees, ut not eveyone knows how to get fom one to the othe. The elationship is given y: C = 9 5 (F 32) whee F is the degees in Fahenheit C is the degee in Celsius Money If you have a pile of quates and dimes, each quate is woth 25 (o $.25) and each dime is woth 10 ($.10), then the value of the pile of coins would e: V =.25q +.10d whee V is the Total Value of money q is the nume of quates d is the nume of dimes 3) Common Geometic Fomulas: Now that you undestand the idea, these ae some asic geometic fomulas that you need to know: l w P = 2l + 2w A = lw P is the peimete l is the length w is the width Rectangle
67 a h Paallelogam P = 2a + 2 A = h P is the peimete a is a side length is the othe side length a h d B Tapezoid P = +a+b+d A = 2 1 h(b+) P is peimete is the shote ase B is the longe ase a is a leg d is a leg h Tiangle P = s1+s2+s3 A = 2 1 h P is the peimete s is a side is the ase a Tiangle c a + + c = 180 a is one angle is anothe angle c is anothe angle
68 SA =2lw+2wh+2lh SA is the Suface Aea H w l Rectangula Solid V = lwh l is the length w is the width Cicle h Cylinde C = 2π A = π 2 LSA = 2πh SA =2πh+2π 2 V = π 2 h V is volume C is the Cicumfeence o Peimete π is a nume, aout 3.14159... it has a utton is the adius of the cicle A is the aea inside the cicle. LSA is Lateal Suface Aea o aea just on the sides π is a nume, aout 3.14159... it has a utton is the adius of the cicle SA is total suface aea V is Volume
69 Cone h l LSA = πl SA = π 2 + πl V = 3 1 π 2 h LSA is Lateal Suface Aea o the aea just on the sides π is a nume, aout 3.14159... it has a utton is the adius of the cicle l is the slant height SA is total suface aea SA = 4π 2 V = 3 4 π 3 SA is the suface aea π is a nume, aout 3.14159... it has a utton Sphee is the adius V is the Volume