Practice Makes Perfect Algebra - Carolyn Wheater [25]
* * *
EXERCISE 10.4
Use the inverse variation equation to find k, and then find the value of the variable requested.
1. If y varies inversely with x and y = 4 when x = 6, find y when x = 8.
2. If y varies inversely with x and y = 8 when x = 4, find x when y = 2.
3. If y varies inversely with x and y = 24 when x = 3, find y when x = 9.
4. If r varies inversely with t and t = 11 when r = 12, find t when r = 3.
5. If a varies inversely with b and a = 54 when b = 10, find a when b = 45.
6. For a fixed area, the length and width of a rectangle are inversely related. When the length is 12 cm, the width is 9 cm. Find the width when the length is 27 cm.
7. At a fixed temperature, the volume of a certain quantity of gas varies inversely with the pressure. If the volume is 16 in3 when the pressure is 18 psi, find the pressure that produces a volume of 20 in3.
8. The time required to drive a fixed distance varies inversely with speed. If a trip takes 6.5 h at 55 mph, how long will it take if driven at 65 mph?
9. The force of gravity on an object varies inversely with its distance from the center of the earth. The radius of the earth is 6378 km. The International Space Station orbits approximately 350 km above earth, and the force of gravity acting on it is approximately 2.7 × 1012 newtons(N). What would the force of gravity be if the Space Station were on the surface of the earth?
10. The illumination on a surface varies inversely with the square of its distance from a light source. At a distance of 5 ft from the light source, the illumination is 270 foot candles(fc). At what distance from the same light source will the illumination be 750 fc?
* * *
Joint and combined variation
A quantity varies jointly with two or more other quantities if it varies directly with their product; that is, y varies jointly with x and z if there is a constant k such that y = kxz. If x is held constant, y varies directly with z. If z is held constant, y varies directly with x.
Combined variation occurs when y varies directly with x and inversely with z. The equation for combined variation has the form . If you hold x constant, y will vary inversely with z. If you hold z constant, y will vary directly with x.
In both joint and combined variation, you can find the value of k just as you did with direct and inverse variation. Choose the correct equation, plug in the known values, and solve for k. Then, just as before, if one or more variables change, plug in the new values along with k and solve for whatever is missing.
* * *
EXERCISE 10.5
Use the appropriate variation equation to find k, and then find the value of the variable requested.
1. If y varies jointly with x and z, and y = 84 when x = 7 and z = 3, find y when x = 9 and z = 5.
2. If y varies jointly with x and z, and y = 165 when x = 6 and z = 11, find x when y = 200 and z = 10.
3. If y varies jointly with x and z, and y = 3375 when x = 3 and z = 9, find z when x = 12 and y = 10,500.
4. If y varies directly with x and inversely with z, and y = 9 when x = 270 and z = 12, find y when x = 100 and z = 8.
5. If y varies directly with x and inversely with z, and y = 53 when x = 3 and z = 12, find x when y = 159 and z = 28.
6. If y varies directly with x and inversely with z, and y = 4 when x = 8 and z = 14, find z when x = 36 and y = 6.
7. The volume of a solid varies jointly with the area of its base and its altitude. When the area of the base is 12 cm2 and the altitude is 16 cm, the volume is 64 cm3. Find the altitude when the volume is 60 cm3 and the area of the base is 9 cm2.
8. The gravitational force between two bodies varies