College Algebra K/DC Monday, 07 March 2016 OBJECTIVE TSW apply (1) direct variation, (2) inverse variation, and (3) joint variation to solve applications. ASSIGNMENT DUE (wire basket) WS Sec. 3.5: Oblique Asymptotes TEST: Sec. 3.4 – 3.6 is on Wed/Thurs, 03/09-10/16. ASSIGNMENT DUE WED/THUR, 03/09-10/16 Sec. 3.5: p. 353 (37-46 all) Sec. 3.6: pp. 365-368 (11-20 all, 21-27 odd, 28-31 all, 33, 35-38 all)

Assignment: Sec. 3.6: pp. 365-368 (11-20 all, 21-27 odd, 28-31 all, 33, 35-38 all) Due on Wed/Thur, 03/09-10/16. Solve each variation problem. 11) If y varies directly as x, and y = 20 when x = 4, find y when x = −6. 12) If y varies directly as x, and y = 9 when x = 30, find y when x = 40. 13) If m varies jointly as x and y, and m = 10 when x = 2 and y = 14, find m when x = 21 and y = 8. 14) If m varies jointly as z and p, and m = 10 when z = 2 and p = 7.5, find m when z = 6 and p = 9. 15) If y varies inversely as x, and y = 10 when x = 3, find y when x = 20.

Assignment: Sec. 3.6: pp. 365-368 (11-20 all, 21-27 odd, 28-31 all, 33, 35-38 all) Due on Wed/Thur, 03/09-10/16. Solve each variation problem. 16) If y varies inversely as x, and y = 20 when x = ¼, find y when x = 15. 17) Suppose r varies directly as the square of m, and inversely as s. If r = 12 when m = 6 and s = 4, find r when m = 6 and s = 20. 18) Suppose p varies directly as the square of z, and inversely as r. If p = 32/5 when z = 4 and r = 10, find p when z = 3 and r = 32. 19) Let a be directly proportional to m and n2, and inversely proportional to y3. If a = 9 when m = 4, n = 9, and y = 3, find a when m = 6, n = 2, and y = 5. 20) If y varies directly as x, and inversely as m2 and r2, and y = 5/3 when x = 1, m = 2, and r = 3, find y when x = 3, m = 1, and r = 8.

Assignment: Sec. 3.6: pp. 365-368 (11-20 all, 21-27 odd, 28-31 all, 33, 35-38 all) Due on Wed/Thur, 03/09-10/16. Solve each variation problem. 21) Circumference of a Circle The circumference of a circle varies directly as the radius. A circle with radius 7 in. has circumference 43.96 in. Find the circumference of the circle if the radius changes to 11 in. 23) Resistance of a Wire The resistance in ohms of a platinum wire temperature sensor varies directly as the temperature in kelvins (K). If the resistance is 646 ohms at a temperature of 190 K, find the resistance at a temperature of 250 K. 25) Distance to the Horizon The distance that a person can see to the horizon on a clear day from a point above the surface of Earth varies directly as the square root of the height at that point. If a person 144 m above the surface of Earth can see 18 km to the horizon, how far can a person see to the horizon from a point 64 m above the surface? 27) Hooke's Law for a Spring Hooke's law for an elastic spring states that the distance a spring stretches varies directly as the force applied. If a force of 15 lb stretches a certain spring 8 in., how much will a force of 30 lb stretch the spring?

Assignment: Sec. 3.6: pp. 365-368 (11-20 all, 21-27 odd, 28-31 all, 33, 35-38 all) Due on Wed/Thur, 03/09-10/16. Solve each variation problem. 28) Current in a Circuit The current in a simple electrical circuit varies inversely as the resistance. If the current is 50 amps when the resistance is 10 ohms, find the current if the resistance is 5 ohms. 29) Speed of a Pulley The speed of a pulley varies inversely as its diameter. One kind of pulley, with diameter 3 in., turns at 150 revolutions per minute. Find the speed of a similar pulley with diameter 5 in. 30) Weight of an Object The weight of an object varies inversely as the square of its distance from the center of Earth. If an object 8000 mi from the center of Earth weighs 90 lb, find its weight when it is 12,000 mi from the center of Earth. 31) Current Flow In electric current flow, it is found that the resistance offered by a fixed length of wire of a given material varies inversely as the square of the diameter of the wire. If a wire 0.01 in. in diameter has a resistance of 0.4 ohm, what is the resistance of a wire of the same length and material with diameter 0.03 in., to the nearest ten-thousandth of an ohm?

Assignment: Sec. 3.6: pp. 365-368 (11-20 all, 21-27 odd, 28-31 all, 33, 35-38 all) Due on Wed/Thur, 03/09-10/16. Solve each variation problem. 33) Simple Interest Simple interest varies jointly as principal and time. If $1000 invested for 2 yr earned $70, find the amount of interest earned by $5000 for 5 yr. 35) Force of Wind The force of the wind blowing on a vertical surface varies jointly as the area of the surface and the square of the velocity. If a wind of 40 mph exerts a force of 50 lb on a surface of ½ ft2, how much force will a wind of 80 mph place on a surface of 2 ft2? 36) Volume of a Cylinder The volume of a right circular cylinder is jointly proportional to the square of the radius of the circular base and to the height. If the volume is 300 cm3 when the height is 10.62 cm and the radius is 3 cm, find the volume to the nearest tenth of a cylinder with radius 4 cm and height 15.92 cm. 37) Sports Arena Construction The roof of a new sports arena rests on round concrete pillars. The maximum load a cylindrical column of circular cross section can hold varies directly as the fourth power of the diameter and inversely as the square of the height. The arena has 9-m- tall columns that are 1 m in diameter and will support a load of 8 metric tons. How many metric tons will be supported by a column 12 m high and 2/3 m in diameter?

Assignment: Sec. 3.6: pp. 365-368 (11-20 all, 21-27 odd, 28-31 all, 33, 35-38 all) Due on Wed/Thur, 03/09-10/16. Solve each variation problem. 38) Sports Arena Construction The sports arena in Exercise 37 requires a horizontal beam 16 m long, 24 cm wide, and 8 cm high. The maximum load of a horizontal beam that is supported at both ends varies directly as the width of the beam and the square of its height and inversely as the length between supports. If a beam of the same material 8 m long, 12 cm wide, and 15 cm high can support a maximum of 400 kg, what is the maximum load the beam in the arena will support?

3.6 Variation Direct Variation ▪ Inverse Variation ▪ Combined Variation ▪ Joint Variation

Direct Variation y varies directly as x (or y is directly proportional to x), if there exists a nonzero real number k, called the constant of variation, such that y = kx. Steps to Solve Variation Problems 1) Write the general relationship (use the constant k). 2) Substitute the given values to find k. 3) Substitute this value for k into the equation. 4) Find the required unknowns and answer the question that is asked.

Direct Variation Ex: If y varies directly as x, and y = 35 when x = 9, find y when x = 7. y = kx 35 = k(9) k = 35/9 y = (35/9)(7) y = 245/9

Direct Variation At a given average speed, the distance traveled by a vehicle varies directly as the time. If a vehicle travels 156 miles in 3 hours, find the distance it will travel in 5 hours at the same average speed. Step 1: Since the distance varies directly as the time, d = kt. Step 2: Substitute d = 156 and t = 3 to find k.

Direct Variation Step 3: The relationship between distance and time is d = 52t. Step 4: Solve the equation for d with t = 5. The vehicle will travel 260 miles in 5 hours. Be sure to properly label all units !!!

Direct Variation The area of a rectangle varies directly as its length. If the area is 50 m2 when the length is 10 m, find the area when the length is 25 m. Step 1: Since the area varies directly as the length, A = kL. Step 2: Substitute A = 50 and L = 10 to find k.

Direct Variation Step 3: The relationship between Area and Length is A = 5L. Step 4: Solve the equation for A with L = 25. The area will be 125 m2 when the length is 25 m.

Inverse Variation Problem Let n be a positive real number. Then y varies inversely as the nth power of x (or y is inversely proportional to the nth power of x), if there exists a nonzero real number k such that If n = 1, then and y varies inversely as x.

Inverse Variation Problem In a certain manufacturing process, the cost of producing a single item varies inversely as the square of the number of items produced. If 100 items are produced, each costs $1.50. Find the cost per item if 250 items are produced. Step 1: Let x represent the number of items produced and y represent the cost per item.

Inverse Variation Problem Step 2: Substitute y = 1.50 and x = 100 to find k. Step 3: The relationship between x and y is Step 4: Solve the equation for y with x = 250. The cost per item will be $0.24.

Combined Direct and Inverse Variation If y varies directly as x and inversely as p and q, and y = 4 when x = −3, p = 2, and q = 5, find y when x = 2, p = 4, and q = 6.

Combined Direct and Inverse Variation If y varies directly as x and inversely as p and q, and y = 4 when x = −3, p = 2, and q = 5, find y when x = 2, p = 4, and q = 6.

Joint Variation Let m and n be real numbers. Then y varies jointly as the nth power of x and the mth power of z if there exists a nonzero real number k such that

Joint Variation Ex: If y varies jointly as the square of x and z, and y = 24 when x = 3 and z = 4, find y when x = 5 and z = 7. y = kx2z 24 = k(3)2(4)  k = 2/3 y = 2/3x2z y = 2/3(5)2(7)  y = 350/3

Inverse Variation Ex: If y varies inversely as the cube of x, and y = 6 when x = 4, find y when x = 2.  k = 384  y = 48

Assignment: Sec. 3.6: pp. 365-368 (11-20 all, 21-27 odd, 28-31 all, 33, 35-38 all) Due on Wed/Thur, 03/09-10/16. Solve each variation problem. 11) If y varies directly as x, and y = 20 when x = 4, find y when x = −6. 12) If y varies directly as x, and y = 9 when x = 30, find y when x = 40. 13) If m varies jointly as x and y, and m = 10 when x = 2 and y = 14, find m when x = 21 and y = 8. 14) If m varies jointly as z and p, and m = 10 when z = 2 and p = 7.5, find m when z = 6 and p = 9. 15) If y varies inversely as x, and y = 10 when x = 3, find y when x = 20.

Assignment: Sec. 3.6: pp. 365-368 (11-20 all, 21-27 odd, 28-31 all, 33, 35-38 all) Due on Wed/Thur, 03/09-10/16. Solve each variation problem. 16) If y varies inversely as x, and y = 20 when x = ¼, find y when x = 15. 17) Suppose r varies directly as the square of m, and inversely as s. If r = 12 when m = 6 and s = 4, find r when m = 6 and s = 20. 18) Suppose p varies directly as the square of z, and inversely as r. If p = 32/5 when z = 4 and r = 10, find p when z = 3 and r = 32. 19) Let a be directly proportional to m and n2, and inversely proportional to y3. If a = 9 when m = 4, n = 9, and y = 3, find a when m = 6, n = 2, and y = 5. 20) If y varies directly as x, and inversely as m2 and r2, and y = 5/3 when x = 1, m = 2, and r = 3, find y when x = 3, m = 1, and r = 8.

Assignment: Sec. 3.6: pp. 365-368 (11-20 all, 21-27 odd, 28-31 all, 33, 35-38 all) Due on Wed/Thur, 03/09-10/16. Solve each variation problem. 21) Circumference of a Circle The circumference of a circle varies directly as the radius. A circle with radius 7 in. has circumference 43.96 in. Find the circumference of the circle if the radius changes to 11 in. 23) Resistance of a Wire The resistance in ohms of a platinum wire temperature sensor varies directly as the temperature in kelvins (K). If the resistance is 646 ohms at a temperature of 190 K, find the resistance at a temperature of 250 K. 25) Distance to the Horizon The distance that a person can see to the horizon on a clear day from a point above the surface of Earth varies directly as the square root of the height at that point. If a person 144 m above the surface of Earth can see 18 km to the horizon, how far can a person see to the horizon from a point 64 m above the surface? 27) Hooke's Law for a Spring Hooke's law for an elastic spring states that the distance a spring stretches varies directly as the force applied. If a force of 15 lb stretches a certain spring 8 in., how much will a force of 30 lb stretch the spring?

Assignment: Sec. 3.6: pp. 365-368 (11-20 all, 21-27 odd, 28-31 all, 33, 35-38 all) Due on Wed/Thur, 03/09-10/16. Solve each variation problem. 28) Current in a Circuit The current in a simple electrical circuit varies inversely as the resistance. If the current is 50 amps when the resistance is 10 ohms, find the current if the resistance is 5 ohms. 29) Speed of a Pulley The speed of a pulley varies inversely as its diameter. One kind of pulley, with diameter 3 in., turns at 150 revolutions per minute. Find the speed of a similar pulley with diameter 5 in. 30) Weight of an Object The weight of an object varies inversely as the square of its distance from the center of Earth. If an object 8000 mi from the center of Earth weighs 90 lb, find its weight when it is 12,000 mi from the center of Earth. 31) Current Flow In electric current flow, it is found that the resistance offered by a fixed length of wire of a given material varies inversely as the square of the diameter of the wire. If a wire 0.01 in. in diameter has a resistance of 0.4 ohm, what is the resistance of a wire of the same length and material with diameter 0.03 in., to the nearest ten-thousandth of an ohm?

Assignment: Sec. 3.6: pp. 365-368 (11-20 all, 21-27 odd, 28-31 all, 33, 35-38 all) Due on Wed/Thur, 03/09-10/16. Solve each variation problem. 33) Simple Interest Simple interest varies jointly as principal and time. If $1000 invested for 2 yr earned $70, find the amount of interest earned by $5000 for 5 yr. 35) Force of Wind The force of the wind blowing on a vertical surface varies jointly as the area of the surface and the square of the velocity. If a wind of 40 mph exerts a force of 50 lb on a surface of ½ ft2, how much force will a wind of 80 mph place on a surface of 2 ft2? 36) Volume of a Cylinder The volume of a right circular cylinder is jointly proportional to the square of the radius of the circular base and to the height. If the volume is 300 cm3 when the height is 10.62 cm and the radius is 3 cm, find the volume to the nearest tenth of a cylinder with radius 4 cm and height 15.92 cm. 37) Sports Arena Construction The roof of a new sports arena rests on round concrete pillars. The maximum load a cylindrical column of circular cross section can hold varies directly as the fourth power of the diameter and inversely as the square of the height. The arena has 9-m- tall columns that are 1 m in diameter and will support a load of 8 metric tons. How many metric tons will be supported by a column 12 m high and 2/3 m in diameter?

Assignment: Sec. 3.6: pp. 365-368 (11-20 all, 21-27 odd, 28-31 all, 33, 35-38 all) Due on Wed/Thur, 03/09-10/16. Solve each variation problem. 38) Sports Arena Construction The sports arena in Exercise 37 requires a horizontal beam 16 m long, 24 cm wide, and 8 cm high. The maximum load of a horizontal beam that is supported at both ends varies directly as the width of the beam and the square of its height and inversely as the length between supports. If a beam of the same material 8 m long, 12 cm wide, and 15 cm high can support a maximum of 400 kg, what is the maximum load the beam in the arena will support?