2/6
M
Variation
“y varies directly with x”
k is the constant of variation
Direct Variation
Step 1: Write generic equation
including k
Step 3: Write the equation using the
specific value for k.
Step 2: Use the given information
to solve for k
Step 4: Use the equation and other
given information to solve for
the other variable(s).
Y varies directly with x. Y is 20 when x is 5.
use to find k
Find y when x is 16.
Y varies directly with x. Y is 8 when x is 12.
use to find k
Find y when x is 27.
use to find y
use to find y
Step 1:
Step 3:
Step 1:
Step 3:
Step 2:
Step 4:
Step 2:
Step 4:
“y varies indirectly with x”
k is the constant of variation
Inverse Variation
Y varies inversely with x. Y is 22.5 when x
is 6.2. Find y when x is 0.1, 0.2, and 0.4.
Y varies inversely with x. Y is 10.2 when x
is 8.5. Find y when x is 2, 3, and 15.
Step 1:
Step 1:
Step 3:
Step 2:
Step 2:
Step 4:
Step 4:
Step 3:
If (x1,y1) and (x2,y2) satisfy xy = k, then x1y1 = x2y2. Find x or y as indicated.
(6,15) and (9,y)
(x,16) and (8,18)
“y varies jointly with x and z”
k is the constant of variation
Joint Variation
Y varies jointly with x and z. Y is -48 when x Y varies jointly with x and z. Y is -28 when x
is 4 and z is 6. Find y when x is 10 and z is 5. is 6 and z is -7. Find y when x is -4 and z is -12.
Step 1:
Step 3:
Step 1:
Step 2:
Step 4:
Step 2:
Step 3:
Step 4:
“y varies directly with x
and inversely with z”
k is the constant of variation
Combined Variation
Y varies directly with x and inversely with z. Y is 2
when x is 3 and z is 6. Find y when x is 10 and z is 8.
Y varies directly with square of x and inversely with the cube
of z. Y is 6 when x is 4 and z is 2. Find y when x is 6 and z is 3.
Step 1:
Step 3:
Step 1:
Step 3:
Step 2:
Step 4:
Step 2:
Step 4:
Heat loss in calories per hour (h), varies jointly with the
difference between the inside and outside temperatures (d),
and the area of the window (A), and inversely as the thickness
of the pane of glass (t). A window with an area of 1200 cm2
and a thickness of 0.4 cm loses 4800 calories per hour when
the temperature difference is 20°F. Find the heat loss for a
window with the same thickness when its area is 2000 cm2
and the temperature difference is 30°F.
Step 1:
Step 3:
Step 2:
Step 4:
The lifting force on the wings of an airplane (F) varies jointly
as the surface area of the wings (A) and the square of the
plane’s airspeed (v). A small plane has a cruising speed of
350 mph. A larger plane is being designed which will require
twice the lifting force on its wings and whose cruising speed
will be 450 mph. How much more surface area will the wings
need to have?
small plane
larger plane
Larger plane needs wings which
have about 20.1% more surface
area than the small plane.