Algebra 2 Preap/GT
Algebra 2 – PreAP/GT
Types of Variation #1
DIRECT VARIATION
y varies directly as x if for a constant k
or
In general, given two ordered pairs and of the same direct variation, the constant
of variation, k, is the same for each equation.
If and ,
then and .
Therefore, .
Example: y varies directly as x.
a) If y = 5 when x = 7.5, find x when y = 9.
b) If y = 5 when x = 7.5, find the equation of variation.
JOINT VARIATION
y varies jointly as x and z if for a constant k
or
In general, given two ordered triples and of the same joint variation, the constant of variation, k, is the same for each equation.
If and ,
then and .
Therefore, .
Example: y varies jointly as x and z.
a) If y = 5 when x = 3 and z = 1.2, find x when y = 4 and z = 6.
b) If y = 5 when x = 3 and z = 1.2, find the equation of variation.
Name ______
INVERSE VARIATION
y varies inversely as x if for a constant k
or
In general, given two ordered pairs and of the same inverse variation, the constant of variation, k, is the same for each equation.
If and ,
then and .
Therefore, .
Example: y varies inversely as x.
a) If y = 3 when x = 10.5, find x when y = 9.
b) If y = 3 when x = 10.5, find the equation of variation.
COMBINED VARIATION
a varies jointly as b and the square of c, and inversely as d if for a constant k
or
In general, given and of the same combinations of variation, the constant of variation, k, is the same for each equation.
If and
then = .
Example: a varies jointly as b and the square of c, and inversely as d.
a) If a = 3 when b = 0.5, c = 2, and d = 6, find d when a = 8, b = 10, and c = 3.
b) a) If a = 3 when b = 0.5, c = 2, and d = 6, find the equation of variation.
1. The distance, d, a spring will stretch varies directly as the force, f, applied. If a force of 20 lbs will stretch a spring 9 in, how far will a force of 35 lbs stretch the spring?
2. The number of British Thermal Units, BTU’s, of heat necessary to hear a building varies inversely as the outside temperature, t. If 500 BTUS’s are required when the outside temperature is 20º F, how many 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 varies jointly with the well depth, d (in ft), and the rate, r (in gal/min). A 55 hp motor can pump water from a depth of 150 ft at a rate of 990 gal/min. 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 support varies directly as the width, w (in inches), of the beam and inversely as the length, l (in feet), of the beam. A beam of length 6 ft and width 1.5 in can support a load of 5 tons. How great of a load can be supported by a beam of the same material that is 4 ft long and 2 in wide?
5. The number of chaperones, c, needed for the class varies directly as the number of students, s, going on the trip. Write the equation of variation if 7 chaperones are needed for 56 students.
6. The cost per box of packing boxes, c, varies inversely with the number of boxes purchased, b. Write the equation of variation if the cost per box is $0.75 when 20 boxes are purchased.
7. The distance, d, Stephen runs each day varies directly with the amount of time, t, that he runs. Find the constant of variation if Stephen runs 10 miles in 1.25 hours.
8. The time, t, that it takes for a group of students to build a sailboat varies inversely as the number of students, n. Find the constant of variation if it took 5 students 195 hours to build a sailboat.
9. The cost, c, of hiring a contractor to build a patio varies jointly as the area, A, in square feet of the patio and the price, p, per square foot of the patio tiles. If the cost is $2832 when the area is 80 sq ft and price of patio tiles is $2.95 per sq ft, find the price per sq ft of the patio tiles when the cost is $4368 and area is 112 sq ft.
10. Joanna’s pay for working overtime, p, varies jointly as the number of hours she works, n, and her hourly pay rate, r. Find the constant of variation if Joanna’s pay for working overtime was $103.44 when n = 8 hours and her hourly pay rate is $8.62.