Algebra 2 Preap/GT

Algebra 2 – PreAP/GT Name ______Types of Variation #1

DIRECT VARIATION INVERSE VARIATION y varies directly as x if for a constant k y varies inversely as x if for a constant k

y k y= kx or = k y = or yx= k x x

In general, given two ordered pairs ( x1, y 1 ) and In general, given two ordered pairs ( x1, y 1) and

( x2, y 2 ) of the same direct variation, the constant ( x2, y 2 ) of the same inverse variation, the of variation, k, is the same for each equation. constant of variation, k, is the same for each equation. If y1= kx 1 and y2= kx 2 , k k If y1 = and y2 = , y y x1 x2 then 1 = k and 2 = k . x1 x2 then y1� x 1 k and y2� x 2 k . y y 1 = 2 Therefore, . y� x y x x1 x 2 Therefore, 1 1 2 2 .

Example: y varies directly as x. Example: y varies inversely as x. a) If y = 5 when x = 7.5, find x when y = 9. a) If y = 3 when x = 10.5, find x when y = 9.

b) If y = 5 when x = 7.5, find the equation of b) If y = 3 when x = 10.5, find the equation of variation. variation.

JOINT VARIATION COMBINED VARIATION y varies jointly as x and z if for a constant k a varies jointly as b and the square of c, and inversely as d if for a constant k y y= kxz or = k kbc2 ad xz a = or = k d bc2 In general, given two ordered triples ( x1, y 1 , z 1 ) and In general, given (a1, b 1 , c 1 , d 1) and ( x2, y 2 , z 2 ) of the same joint variation, the constant of variation, k, is the same for each equation. (a2, b 2 , c 2 , d 2 ) of the same combinations of variation, the constant of variation, k, is the same If y1= k贩 x 1 z 1 and y2= k贩 x 2 z 2 , for each equation.

y y a1 d 1 a2 d 2 1 = k 2 = k If = k and = k then and . b c 2 b c 2 x1 z 1 x2 z 2 1 1 2 2

a d a d y1 y 2 1 1 2 2 Therefore, = . then 2 = 2 . x1贩 z 1 x 2 z 2 b1 c 1 b2 c 2

Example: y varies jointly as x and z. Example: a varies jointly as b and the square of c, and inversely as d. a) If y = 5 when x = 3 and z = 1.2, find x when y = 4 and z = 6. a) If a = 3 when b = 0.5, c = 2, and d = 6, find d when a = 8, b = 10, and c = 3. b) If y = 5 when x = 3 and z = 1.2, find the equation b) a) If a = 3 when b = 0.5, c = 2, and d = 6, find of variation. the equation of variation. 1. The distance, d, a spring will stretch varies directly 6. The cost per box of packing boxes, c, varies as the force, f, applied. If a force of 20 lbs will inversely with the number of boxes purchased, b. stretch a spring 9 in, how far will a force of 35 lbs Write the equation of variation if the cost per box is stretch the spring? $0.75 when 20 boxes are purchased.

2. The number of British Thermal Units, BTU’s, of 7. The distance, d, Stephen runs each day varies heat necessary to hear a building varies inversely as directly with the amount of time, t, that he runs. the outside temperature, t. If 500 BTUS’s are required Find the constant of variation if Stephen runs 10 when the outside temperature is 20º F, how many miles in 1.25 hours. BTU’s will be required to heat the building if the outside temperature is 4º F?

3. The horsepower, h, needed for a water pump 8. The time, t, that it takes for a group of students varies jointly with the well depth, d (in ft), and the to build a sailboat varies inversely as the number of rate, r (in gal/min). A 55 hp motor can pump water students, n. Find the constant of variation if it took from a depth of 150 ft at a rate of 990 gal/min. 5 students 195 hours to build a sailboat. Estimate the horsepower (to the nearest hp) of a solar energy pump if it can pump water from a depth of 1000 ft at a rate of 2.75 gal/min.

4. The load, ld, that a beam of constant depth can 9. The cost, c, of hiring a contractor to build a patio support varies directly as the width, w (in inches), of varies jointly as the area, A, in square feet of the the beam and inversely as the length, l (in feet), of the patio and the price, p, per square foot of the patio beam. A beam of length 6 ft and width 1.5 in can tiles. If the cost is $2832 when the area is 80 sq ft support a load of 5 tons. How great of a load can be and price of patio tiles is $2.95 per sq ft, find the supported by a beam of the same material that is 4 ft price per sq ft of the patio tiles when the cost is long and 2 in wide? $4368 and area is 112 sq ft.

5. The number of chaperones, c, needed for the class 10. Joanna’s pay for working overtime, p, varies varies directly as the number of students, s, going on jointly as the number of hours she works, n, and her the trip. Write the equation of variation if 7 hourly pay rate, r. Find the constant of variation if chaperones are needed for 56 students. Joanna’s pay for working overtime was $103.44 when n = 8 hours and her hourly pay rate is $8.62.